geometry problems from Online Math Open Olympiads (OMO)
with aops links in the names
2012 OMO Winter p5
Congruent circles $\Gamma_1$ and $\Gamma_2$ have radius $2012,$ and the center of $\Gamma_1$ lies on $\Gamma_2.$ Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. The line through $A$ perpendicular to $AB$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$, respectively. Find the length of $CD$.
A cross-pentomino is a shape that consists of a unit square and four other unit squares each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R,$ what is $100R^2$?
A circle $\omega$ has center $O$ and radius $r$. A chord $BC$ of $\omega$ also has length $r$, and the tangents to $\omega$ at $B$ and $C$ meet at $A$. Ray $AO$ meets $\omega$ at $D$ past $O$, and ray $OA$ meets the circle centered at $A$ with radius $AB$ at $E$ past $A$. Compute the degree measure of $\angle DBE$.
2012 OMO Fall p5
Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?
In acute triangle $ABC$ let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Determine the value of $CE^2$.
Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$.
Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$.
In trapezoid $ABCD$, $AB < CD$, $AB\perp BC$, $AB\parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD\perp DC$. If $\frac{QP} {AP}+\frac{AP} {QP} = \left( \frac{51}{14}\right)^4 - 2,$ then $\frac{BP} {AP}-\frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$.
In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ (not containing $A$) of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the point on the line $MI_a$ such that $AR\parallel DP$. Given that $\frac{AI_a}{AI}=9$, the ratio $\frac{QM} {RI_a}$ can be expressed in the form $\frac{m}{n}$ for two relatively prime positive integers $m,n$. Compute $m+n$.
Let $ABC$ be a triangle with circumcircle $\omega$. Let the bisector of $\angle ABC$ meet segment $AC$ at $D$ and circle $\omega$ at $M\ne B$. The circumcircle of $\triangle BDC$ meets line $AB$ at $E\ne B$, and $CE$ meets $\omega$ at $P\ne C$. The bisector of $\angle PMC$ meets segment $AC$ at $Q\ne C$. Given that $PQ = MC$, determine the degree measure of $\angle ABC$.
Three lines $m$, $n$, and $\ell$ lie in a plane such that no two are parallel. Lines $m$ and $n$ meet at an acute angle of $14^{\circ}$, and lines $m$ and $\ell$ meet at an acute angle of $20^{\circ}$. Find, in degrees, the sum of all possible acute angles formed by lines $n$ and $\ell$.
2014 OMO Spring p2
Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$?
http://internetolympiad.org/
with aops links in the names
2012 - 2020
(Spring only in 2020)
(Spring only in 2020)
2012 OMO Winter p5
Congruent circles $\Gamma_1$ and $\Gamma_2$ have radius $2012,$ and the center of $\Gamma_1$ lies on $\Gamma_2.$ Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. The line through $A$ perpendicular to $AB$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$, respectively. Find the length of $CD$.
Ray Li
2012 OMO Winter p12A cross-pentomino is a shape that consists of a unit square and four other unit squares each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R,$ what is $100R^2$?
Ray Li
2012 OMO Winter p13A circle $\omega$ has center $O$ and radius $r$. A chord $BC$ of $\omega$ also has length $r$, and the tangents to $\omega$ at $B$ and $C$ meet at $A$. Ray $AO$ meets $\omega$ at $D$ past $O$, and ray $OA$ meets the circle centered at $A$ with radius $AB$ at $E$ past $A$. Compute the degree measure of $\angle DBE$.
Ray Li
2012 OMO Winter p16
Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$. Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b$.
Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$. Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b$.
Alex Zhu
2012 OMO Winter p20
Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle?
Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle?
Ray Li
2012 OMO Winter p23
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Ray Li
2012 OMO Winter p31
Let $ABC$ be a triangle inscribed in circle $\Gamma$, centered at $O$ with radius $333.$ Let $M$ be the midpoint of $AB$, $N$ be the midpoint of $AC$, and $D$ be the point where line $AO$ intersects $BC$. Given that lines $MN$ and $BO$ concur on $\Gamma$ and that $BC = 665$, find the length of segment $AD$.
Let $ABC$ be a triangle inscribed in circle $\Gamma$, centered at $O$ with radius $333.$ Let $M$ be the midpoint of $AB$, $N$ be the midpoint of $AC$, and $D$ be the point where line $AO$ intersects $BC$. Given that lines $MN$ and $BO$ concur on $\Gamma$ and that $BC = 665$, find the length of segment $AD$.
Alex Zhu
2012 OMO Winter p37
In triangle $ABC$, $AB = 1$ and $AC = 2$. Suppose there exists a point $P$ in the interior of triangle $ABC$ such that $\angle PBC = 70^{\circ}$, and that there are points $E$ and $D$ on segments $AB$ and $AC$, such that $\angle BPE = \angle EPA = 75^{\circ}$ and $\angle APD = \angle DPC = 60^{\circ}$. Let $BD$ meet $CE$ at $Q,$ and let $AQ$ meet $BC$ at $F.$ If $M$ is the midpoint of $BC$, compute the degree measure of $\angle MPF.$
In triangle $ABC$, $AB = 1$ and $AC = 2$. Suppose there exists a point $P$ in the interior of triangle $ABC$ such that $\angle PBC = 70^{\circ}$, and that there are points $E$ and $D$ on segments $AB$ and $AC$, such that $\angle BPE = \angle EPA = 75^{\circ}$ and $\angle APD = \angle DPC = 60^{\circ}$. Let $BD$ meet $CE$ at $Q,$ and let $AQ$ meet $BC$ at $F.$ If $M$ is the midpoint of $BC$, compute the degree measure of $\angle MPF.$
Alex Zhu and Ray Li
2012 OMO Winter p42
In triangle $ABC,$ $\sin \angle A=\frac{4}{5}$ and $\angle A<90^\circ$ Let $D$ be a point outside triangle $ABC$ such that $\angle BAD=\angle DAC$ and $\angle BDC = 90^{\circ}.$ Suppose that $AD=1$ and that $\frac{BD} {CD} = \frac{3}{2}.$ If $AB+AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are pairwise relatively prime integers, find $a+b+c$.
In triangle $ABC,$ $\sin \angle A=\frac{4}{5}$ and $\angle A<90^\circ$ Let $D$ be a point outside triangle $ABC$ such that $\angle BAD=\angle DAC$ and $\angle BDC = 90^{\circ}.$ Suppose that $AD=1$ and that $\frac{BD} {CD} = \frac{3}{2}.$ If $AB+AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are pairwise relatively prime integers, find $a+b+c$.
Ray Li
2012 OMO Winter p47
Let $ABCD$ be an isosceles trapezoid with bases $AB=5$ and $CD=7$ and legs $BC=AD=2 \sqrt{10}.$ A circle $\omega$ with center $O$ passes through $A,B,C,$ and $D.$ Let $M$ be the midpoint of segment $CD,$ and ray $AM$ meet $\omega$ again at $E.$ Let $N$ be the midpoint of $BE$ and $P$ be the intersection of $BE$ with $CD.$ Let $Q$ be the intersection of ray $ON$ with ray $DC.$ There is a point $R$ on the circumcircle of $PNQ$ such that $\angle PRC = 45^\circ.$ The length of $DR$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Let $ABCD$ be an isosceles trapezoid with bases $AB=5$ and $CD=7$ and legs $BC=AD=2 \sqrt{10}.$ A circle $\omega$ with center $O$ passes through $A,B,C,$ and $D.$ Let $M$ be the midpoint of segment $CD,$ and ray $AM$ meet $\omega$ again at $E.$ Let $N$ be the midpoint of $BE$ and $P$ be the intersection of $BE$ with $CD.$ Let $Q$ be the intersection of ray $ON$ with ray $DC.$ There is a point $R$ on the circumcircle of $PNQ$ such that $\angle PRC = 45^\circ.$ The length of $DR$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Ray Li
2012 OMO Winter p50
In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be the largest possible value of the circumradius of face $ABC$. Given that $R$ can be expressed in the form $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be the largest possible value of the circumradius of face $ABC$. Given that $R$ can be expressed in the form $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
Alex Zhu
Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?
Ray Li
2012 OMO Fall p8In acute triangle $ABC$ let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Determine the value of $CE^2$.
Ray Li
2012 OMO Fall p11Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$.
Ray Li
2012 OMO Fall p16Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$.
Ray Li
2012 OMO Fall p19In trapezoid $ABCD$, $AB < CD$, $AB\perp BC$, $AB\parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD\perp DC$. If $\frac{QP} {AP}+\frac{AP} {QP} = \left( \frac{51}{14}\right)^4 - 2,$ then $\frac{BP} {AP}-\frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$.
Ray Li
2012 OMO Fall p24In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ (not containing $A$) of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the point on the line $MI_a$ such that $AR\parallel DP$. Given that $\frac{AI_a}{AI}=9$, the ratio $\frac{QM} {RI_a}$ can be expressed in the form $\frac{m}{n}$ for two relatively prime positive integers $m,n$. Compute $m+n$.
Ray Li
2012 OMO Fall p27Let $ABC$ be a triangle with circumcircle $\omega$. Let the bisector of $\angle ABC$ meet segment $AC$ at $D$ and circle $\omega$ at $M\ne B$. The circumcircle of $\triangle BDC$ meets line $AB$ at $E\ne B$, and $CE$ meets $\omega$ at $P\ne C$. The bisector of $\angle PMC$ meets segment $AC$ at $Q\ne C$. Given that $PQ = MC$, determine the degree measure of $\angle ABC$.
Ray Li
2013 OMO Winter p3Three lines $m$, $n$, and $\ell$ lie in a plane such that no two are parallel. Lines $m$ and $n$ meet at an acute angle of $14^{\circ}$, and lines $m$ and $\ell$ meet at an acute angle of $20^{\circ}$. Find, in degrees, the sum of all possible acute angles formed by lines $n$ and $\ell$.
Ray Li
2013 OMO Winter p6
Circle $S_1$ has radius $5$. Circle $S_2$ has radius $7$ and has its center lying on $S_1$. Circle $S_3$ has an integer radius and has its center lying on $S_2$. If the center of $S_1$ lies on $S_3$, how many possible values are there for the radius of $S_3$?
Circle $S_1$ has radius $5$. Circle $S_2$ has radius $7$ and has its center lying on $S_1$. Circle $S_3$ has an integer radius and has its center lying on $S_2$. If the center of $S_1$ lies on $S_3$, how many possible values are there for the radius of $S_3$?
Ray Li
2013 OMO Winter p11
Let $A$, $B$, and $C$ be distinct points on a line with $AB=AC=1$. Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$. What is the degree measure of the acute angle formed by lines $EC$ and $BF$?
Let $A$, $B$, and $C$ be distinct points on a line with $AB=AC=1$. Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$. What is the degree measure of the acute angle formed by lines $EC$ and $BF$?
Ray Li
2013 OMO Winter p16
Let $S_1$ and $S_2$ be two circles intersecting at points $A$ and $B$. Let $C$ and $D$ be points on $S_1$ and $S_2$ respectively such that line $CD$ is tangent to both circles and $A$ is closer to line $CD$ than $B$. If $\angle BCA = 52^\circ$ and $\angle BDA = 32^\circ$, determine the degree measure of $\angle CBD$.
Let $S_1$ and $S_2$ be two circles intersecting at points $A$ and $B$. Let $C$ and $D$ be points on $S_1$ and $S_2$ respectively such that line $CD$ is tangent to both circles and $A$ is closer to line $CD$ than $B$. If $\angle BCA = 52^\circ$ and $\angle BDA = 32^\circ$, determine the degree measure of $\angle CBD$.
Ray Li
2013 OMO Winter p19
$A,B,C$ are points in the plane such that $\angle ABC=90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA=20$ and $BC=21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$A,B,C$ are points in the plane such that $\angle ABC=90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA=20$ and $BC=21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Ray Li
2013 OMO Winter p22
In triangle $ABC$, $AB = 28$, $AC = 36$, and $BC = 32$. Let $D$ be the point on segment $BC$ satisfying $\angle BAD = \angle DAC$, and let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $ABC$. Find the length of segment $AE$.
In triangle $ABC$, $AB = 28$, $AC = 36$, and $BC = 32$. Let $D$ be the point on segment $BC$ satisfying $\angle BAD = \angle DAC$, and let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $ABC$. Find the length of segment $AE$.
Ray Li
2013 OMO Winter p26
In triangle $ABC$, $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD,BE$ are perpendicular to $CF$. $M$ is the midpoint of $AB$. If $ME=13$, $AD=15$, and $BE=25$, find $AC+CB$.
In triangle $ABC$, $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD,BE$ are perpendicular to $CF$. $M$ is the midpoint of $AB$. If $ME=13$, $AD=15$, and $BE=25$, find $AC+CB$.
Ray Li
2013 OMO Winter p32
In $\triangle ABC$ with incenter $I$, $AB = 61$, $AC = 51$, and $BC=71$. The circumcircles of triangles $AIB$ and $AIC$ meet line $BC$ at points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Determine the length of segment $DE$.
In $\triangle ABC$ with incenter $I$, $AB = 61$, $AC = 51$, and $BC=71$. The circumcircles of triangles $AIB$ and $AIC$ meet line $BC$ at points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Determine the length of segment $DE$.
James Tao
2013 OMO Winter p36
Let $ABCD$ be a nondegenerate isosceles trapezoid with integer side lengths such that $BC \parallel AD$ and $AB=BC=CD$. Given that the distance between the incenters of triangles $ABD$ and $ACD$ is $8!$, determine the number of possible lengths of segment $AD$.
Let $ABCD$ be a nondegenerate isosceles trapezoid with integer side lengths such that $BC \parallel AD$ and $AB=BC=CD$. Given that the distance between the incenters of triangles $ABD$ and $ACD$ is $8!$, determine the number of possible lengths of segment $AD$.
Ray Li
2013 OMO Winter p38
Triangle $ABC$ has sides $AB = 25$, $BC = 30$, and $CA=20$. Let $P,Q$ be the points on segments $AB,AC$, respectively, such that $AP=5$ and $AQ=4$. Suppose lines $BQ$ and $CP$ intersect at $R$ and the circumcircles of $\triangle{BPR}$ and $\triangle{CQR}$ intersect at a second point $S\ne R$. If the length of segment $SA$ can be expressed in the form $\frac{m}{\sqrt{n}}$ for positive integers $m,n$, where $n$ is not divisible by the square of any prime, find $m+n$.
Triangle $ABC$ has sides $AB = 25$, $BC = 30$, and $CA=20$. Let $P,Q$ be the points on segments $AB,AC$, respectively, such that $AP=5$ and $AQ=4$. Suppose lines $BQ$ and $CP$ intersect at $R$ and the circumcircles of $\triangle{BPR}$ and $\triangle{CQR}$ intersect at a second point $S\ne R$. If the length of segment $SA$ can be expressed in the form $\frac{m}{\sqrt{n}}$ for positive integers $m,n$, where $n$ is not divisible by the square of any prime, find $m+n$.
Victor Wang
2013 OMO Winter p40
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Let $M$ be the midpoint of $BC$ and let $\Gamma$ be the circle passing through $A$ and tangent to line $BC$ at $M$. Let $\Gamma$ intersect lines $AB$ and $AC$ at points $D$ and $E$, respectively, and let $N$ be the midpoint of $DE$. Suppose line $MN$ intersects lines $AB$ and $AC$ at points $P$ and $O$, respectively. If the ratio $MN:NO:OP$ can be written in the form $a:b:c$ with $a,b,c$ positive integers satisfying $\gcd(a,b,c)=1$, find $a+b+c$.
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Let $M$ be the midpoint of $BC$ and let $\Gamma$ be the circle passing through $A$ and tangent to line $BC$ at $M$. Let $\Gamma$ intersect lines $AB$ and $AC$ at points $D$ and $E$, respectively, and let $N$ be the midpoint of $DE$. Suppose line $MN$ intersects lines $AB$ and $AC$ at points $P$ and $O$, respectively. If the ratio $MN:NO:OP$ can be written in the form $a:b:c$ with $a,b,c$ positive integers satisfying $\gcd(a,b,c)=1$, find $a+b+c$.
James Tao
2013 OMO Winter p44
Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$.
Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$.
Ray Li
2013 OMO Winter p46
Let $ABC$ be a triangle with $\angle B - \angle C = 30^{\circ}$. Let $D$ be the point where the $A$-excircle touches line $BC$, $O$ the circumcenter of triangle $ABC$, and $X,Y$ the intersections of the altitude from $A$ with the incircle with $X$ in between $A$ and $Y$. Suppose points $A$, $O$ and $D$ are collinear. If the ratio $\frac{AO}{AX}$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $\gcd(a,b,d)=1$ and $c$ not divisible by the square of any prime, find $a+b+c+d$.
Let $ABC$ be a triangle with $\angle B - \angle C = 30^{\circ}$. Let $D$ be the point where the $A$-excircle touches line $BC$, $O$ the circumcenter of triangle $ABC$, and $X,Y$ the intersections of the altitude from $A$ with the incircle with $X$ in between $A$ and $Y$. Suppose points $A$, $O$ and $D$ are collinear. If the ratio $\frac{AO}{AX}$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $\gcd(a,b,d)=1$ and $c$ not divisible by the square of any prime, find $a+b+c+d$.
James Tao
2013 OMO Winter p49
In $\triangle ABC$, $CA=1960\sqrt{2}$, $CB=6720$, and $\angle C = 45^{\circ}$. Let $K$, $L$, $M$ lie on line $BC$, $CA$, and $AB$ such that $AK \perp BC$, $BL \perp CA$, and $AM=BM$. Let $N$, $O$, $P$ lie on line $KL$, $BA$, and $BL$ such that $AN=KN$, $BO=CO$, and $A$ lies on line $NP$. If $H$ is the orthocenter of $\triangle MOP$, compute $HK^2$.
In $\triangle ABC$, $CA=1960\sqrt{2}$, $CB=6720$, and $\angle C = 45^{\circ}$. Let $K$, $L$, $M$ lie on line $BC$, $CA$, and $AB$ such that $AK \perp BC$, $BL \perp CA$, and $AM=BM$. Let $N$, $O$, $P$ lie on line $KL$, $BA$, and $BL$ such that $AN=KN$, $BO=CO$, and $A$ lies on line $NP$. If $H$ is the orthocenter of $\triangle MOP$, compute $HK^2$.
Evan Chen
2013 OMO Fall p7
Points $M$, $N$, $P$ are selected on sides $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, respectively, of triangle $ABC$. Find the area of triangle $MNP$ given that $AM=MB=BP=15$ and $AN=NC=CP=25$.
Points $M$, $N$, $P$ are selected on sides $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, respectively, of triangle $ABC$. Find the area of triangle $MNP$ given that $AM=MB=BP=15$ and $AN=NC=CP=25$.
Evan Chen
2013 OMO Fall p9
Let $AXYZB$ be a regular pentagon with area $5$ inscribed in a circle with center $O$. Let $Y'$ denote the reflection of $Y$ over $\overline{AB}$ and suppose $C$ is the center of a circle passing through $A$, $Y'$ and $B$. Compute the area of triangle $ABC$.
Let $AXYZB$ be a regular pentagon with area $5$ inscribed in a circle with center $O$. Let $Y'$ denote the reflection of $Y$ over $\overline{AB}$ and suppose $C$ is the center of a circle passing through $A$, $Y'$ and $B$. Compute the area of triangle $ABC$.
Evan Chen
2013 OMO Fall p10
In convex quadrilateral $AEBC$, $\angle BEA = \angle CAE = 90^{\circ}$ and $AB = 15$, $BC = 14$ and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to $\overline{AB}$. If ray $CD$ meets $\overline{AE}$ at $F$, compute $AE \cdot AF$.
In convex quadrilateral $AEBC$, $\angle BEA = \angle CAE = 90^{\circ}$ and $AB = 15$, $BC = 14$ and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to $\overline{AB}$. If ray $CD$ meets $\overline{AE}$ at $F$, compute $AE \cdot AF$.
David Stoner
2013 OMO Fall p17
Let $ABXC$ be a parallelogram. Points $K,P,Q$ lie on $\overline{BC}$ in this order such that $BK = \frac{1}{3} KC$ and $BP = PQ = QC = \frac{1}{3} BC$. Rays $XP$ and $XQ$ meet $\overline{AB}$ and $\overline{AC}$ at $D$ and $E$, respectively. Suppose that $\overline{AK} \perp \overline{BC}$, $EK-DK=9$ and $BC=60$. Find $AB+AC$.
Let $ABXC$ be a parallelogram. Points $K,P,Q$ lie on $\overline{BC}$ in this order such that $BK = \frac{1}{3} KC$ and $BP = PQ = QC = \frac{1}{3} BC$. Rays $XP$ and $XQ$ meet $\overline{AB}$ and $\overline{AC}$ at $D$ and $E$, respectively. Suppose that $\overline{AK} \perp \overline{BC}$, $EK-DK=9$ and $BC=60$. Find $AB+AC$.
Evan Chen
2013 OMO Fall p21
Let $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$. Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
Let $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$. Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
Ray Li
2013 OMO Fall p23
Let $ABCDE$ be a regular pentagon, and let $F$ be a point on $\overline{AB}$ with $\angle CDF=55^\circ$. Suppose $\overline{FC}$ and $\overline{BE}$ meet at $G$, and select $H$ on the extension of $\overline{CE}$ past $E$ such that $\angle DHE=\angle FDG$. Find the measure of $\angle GHD$, in degrees.
Let $ABCDE$ be a regular pentagon, and let $F$ be a point on $\overline{AB}$ with $\angle CDF=55^\circ$. Suppose $\overline{FC}$ and $\overline{BE}$ meet at $G$, and select $H$ on the extension of $\overline{CE}$ past $E$ such that $\angle DHE=\angle FDG$. Find the measure of $\angle GHD$, in degrees.
David Stoner
2013 OMO Fall p25
Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.
Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.
Evan Chen
2013 OMO Fall p26
Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$.
Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$.
Michael Kural
Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$?
Evan Chen
Let $A_1A_2 \dots A_{4000}$ be a regular $4000$-gon. Let $X$ be the foot of the altitude from $A_{1986}$ onto diagonal $A_{1000}A_{3000}$, and let $Y$ be the foot of the altitude from $A_{2014}$ onto $A_{2000}A_{4000}$. If $XY = 1$, what is the area of square $A_{500}A_{1500}A_{2500}A_{3500}$?
Evan Chen
2014 OMO Spring p11
Let $X$ be a point inside convex quadrilateral $ABCD$ with $\angle AXB+\angle CXD=180^{\circ}$. If $AX=14$, $BX=11$, $CX=5$, $DX=10$, and $AB=CD$, find the sum of the areas of $\triangle AXB$ and $\triangle CXD$.
Let $ABC$ be an isosceles triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ are selected on sides $AB$ and $AC$, and points $X$ and $Y$ are the feet of the altitudes from $D$ and $E$ to side $BC$. Given that $AD = 48\sqrt2$ and $AE = 52\sqrt2$, compute $XY$.Let $X$ be a point inside convex quadrilateral $ABCD$ with $\angle AXB+\angle CXD=180^{\circ}$. If $AX=14$, $BX=11$, $CX=5$, $DX=10$, and $AB=CD$, find the sum of the areas of $\triangle AXB$ and $\triangle CXD$.
Michael Kural
2014 OMO Spring p12
The points $A$, $B$, $C$, $D$, $E$ lie on a line $\ell$ in this order. Suppose $T$ is a point not on $\ell$ such that $\angle BTC = \angle DTE$, and $\overline{AT}$ is tangent to the circumcircle of triangle $BTE$. If $AB = 2$, $BC = 36$, and $CD = 15$, compute $DE$.
The points $A$, $B$, $C$, $D$, $E$ lie on a line $\ell$ in this order. Suppose $T$ is a point not on $\ell$ such that $\angle BTC = \angle DTE$, and $\overline{AT}$ is tangent to the circumcircle of triangle $BTE$. If $AB = 2$, $BC = 36$, and $CD = 15$, compute $DE$.
Yang Liu
2014 OMO Spring p14
Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$, $AC = 1800$, $BC = 2014$. The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$. Compute the length $XY$.
Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$, $AC = 1800$, $BC = 2014$. The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$. Compute the length $XY$.
Evan Chen
2014 OMO Spring p17
Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$.
Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$.
Evan Chen
2014 OMO Spring p20
Let $ABC$ be an acute triangle with circumcenter $O$, and select $E$ on $\overline{AC}$ and $F$ on $\overline{AB}$ so that $\overline{BE} \perp \overline{AC}$, $\overline{CF} \perp \overline{AB}$. Suppose $\angle EOF - \angle A = 90^{\circ}$ and $\angle AOB - \angle B = 30^{\circ}$. If the maximum possible measure of $\angle C$ is $\tfrac mn \cdot 180^{\circ}$ for some positive integers $m$ and $n$ with $m < n$ and $\gcd(m,n)=1$, compute $m+n$.
Let $ABC$ be an acute triangle with circumcenter $O$, and select $E$ on $\overline{AC}$ and $F$ on $\overline{AB}$ so that $\overline{BE} \perp \overline{AC}$, $\overline{CF} \perp \overline{AB}$. Suppose $\angle EOF - \angle A = 90^{\circ}$ and $\angle AOB - \angle B = 30^{\circ}$. If the maximum possible measure of $\angle C$ is $\tfrac mn \cdot 180^{\circ}$ for some positive integers $m$ and $n$ with $m < n$ and $\gcd(m,n)=1$, compute $m+n$.
Evan Chen
2014 OMO Spring p23
Let $\Gamma_1$ and $\Gamma_2$ be circles in the plane with centers $O_1$ and $O_2$ and radii $13$ and $10$, respectively. Assume $O_1O_2=2$. Fix a circle $\Omega$ with radius $2$, internally tangent to $\Gamma_1$ at $P$ and externally tangent to $\Gamma_2$ at $Q$ . Let $\omega$ be a second variable circle internally tangent to $\Gamma_1$ at $X$ and externally tangent to $\Gamma_2$ at $Y$. Line $PQ$ meets $\Gamma_2$ again at $R$, line $XY$ meets $\Gamma_2$ again at $Z$, and lines $PZ$ and $XR$ meet at $M$.
As $\omega$ varies, the locus of point $M$ encloses a region of area $\tfrac{p}{q} \pi$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.
Let $\Gamma_1$ and $\Gamma_2$ be circles in the plane with centers $O_1$ and $O_2$ and radii $13$ and $10$, respectively. Assume $O_1O_2=2$. Fix a circle $\Omega$ with radius $2$, internally tangent to $\Gamma_1$ at $P$ and externally tangent to $\Gamma_2$ at $Q$ . Let $\omega$ be a second variable circle internally tangent to $\Gamma_1$ at $X$ and externally tangent to $\Gamma_2$ at $Y$. Line $PQ$ meets $\Gamma_2$ again at $R$, line $XY$ meets $\Gamma_2$ again at $Z$, and lines $PZ$ and $XR$ meet at $M$.
As $\omega$ varies, the locus of point $M$ encloses a region of area $\tfrac{p}{q} \pi$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.
Michael Kural
2014 OMO Spring p29
Let $ABCD$ be a tetrahedron whose six side lengths are all integers, and let $N$ denote the sum of these side lengths. There exists a point $P$ inside $ABCD$ such that the feet from $P$ onto the faces of the tetrahedron are the orthocenter of $\triangle ABC$, centroid of $\triangle BCD$, circumcenter of $\triangle CDA$, and orthocenter of $\triangle DAB$. If $CD = 3$ and $N < 100{,}000$, determine the maximum possible value of $N$.
Let $ABCD$ be a tetrahedron whose six side lengths are all integers, and let $N$ denote the sum of these side lengths. There exists a point $P$ inside $ABCD$ such that the feet from $P$ onto the faces of the tetrahedron are the orthocenter of $\triangle ABC$, centroid of $\triangle BCD$, circumcenter of $\triangle CDA$, and orthocenter of $\triangle DAB$. If $CD = 3$ and $N < 100{,}000$, determine the maximum possible value of $N$.
Sammy Luo and Evan Chen
2014 OMO Fall p1
Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by $x$ square units. What is the smallest possible positive value of $x$?
Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by $x$ square units. What is the smallest possible positive value of $x$?
Ray Li
2014 OMO Fall p3
Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$, we color red the foot of the perpendicular from $C$ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$ (that is, find the greatest integer not exceeding $\mathcal A$).
Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$, we color red the foot of the perpendicular from $C$ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$ (that is, find the greatest integer not exceeding $\mathcal A$).
Yang Liu
2014 OMO Fall p4
A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom 52 = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$.
How many different triangles can Tina draw? (Similar triangles are considered the same.)
A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom 52 = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$.
Yang Liu
For an olympiad geometry problem, Tina wants to draw an acute triangle whose angles each measure a multiple of $10^{\circ}$. She doesn't want her triangle to have any special properties, so none of the angles can measure $30^{\circ}$ or $60^{\circ}$, and the triangle should definitely not be isosceles.How many different triangles can Tina draw? (Similar triangles are considered the same.)
Evan Chen
2014 OMO Fall p11
Given a triangle $ABC$, consider the semicircle with diameter $\overline{EF}$ on $\overline{BC}$ tangent to $\overline{AB}$ and $\overline{AC}$. If $BE=1$, $EF=24$, and $FC=3$, find the perimeter of $\triangle{ABC}$.
Given a triangle $ABC$, consider the semicircle with diameter $\overline{EF}$ on $\overline{BC}$ tangent to $\overline{AB}$ and $\overline{AC}$. If $BE=1$, $EF=24$, and $FC=3$, find the perimeter of $\triangle{ABC}$.
Proposed by Ray Li
2014 OMO Fall p16
Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.)
Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.)
Robin Park
2014 OMO Fall p17
Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$
Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$
Ray Li
2014 OMO Fall p19
In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos ^2 \angle AEI = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, determine $m+n$.
In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos ^2 \angle AEI = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, determine $m+n$.
Ray Li
2014 OMO Fall p20
Let $n = 2188 = 3^7+1$ and let $A_0^{(0)}, A_1^{(0)}, ..., A_{n-1}^{(0)}$ be the vertices of a regular $n$-gon (in that order) with center $O$ . For $i = 1, 2, \dots, 7$ and $j=0,1,\dots,n-1$, let $A_j^{(i)}$ denote the centroid of the triangle $ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)}. $Here the subscripts are taken modulo $n$. If \[ \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \] for relatively prime positive integers $p$ and $q$, find $p+q$.
Let $n = 2188 = 3^7+1$ and let $A_0^{(0)}, A_1^{(0)}, ..., A_{n-1}^{(0)}$ be the vertices of a regular $n$-gon (in that order) with center $O$ . For $i = 1, 2, \dots, 7$ and $j=0,1,\dots,n-1$, let $A_j^{(i)}$ denote the centroid of the triangle $ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)}. $Here the subscripts are taken modulo $n$. If \[ \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \] for relatively prime positive integers $p$ and $q$, find $p+q$.
Yang Liu
2014 OMO Fall p26
Let $ABC$ be a triangle with $AB=26$, $AC=28$, $BC=30$. Let $X$, $Y$, $Z$ be the midpoints of arcs $BC$, $CA$, $AB$ (not containing the opposite vertices) respectively on the circumcircle of $ABC$. Let $P$ be the midpoint of arc $BC$ containing point $A$. Suppose lines $BP$ and $XZ$ meet at $M$ , while lines $CP$ and $XY$ meet at $N$. Find the square of the distance from $X$ to $MN$.
Let $ABC$ be a triangle with $AB=26$, $AC=28$, $BC=30$. Let $X$, $Y$, $Z$ be the midpoints of arcs $BC$, $CA$, $AB$ (not containing the opposite vertices) respectively on the circumcircle of $ABC$. Let $P$ be the midpoint of arc $BC$ containing point $A$. Suppose lines $BP$ and $XZ$ meet at $M$ , while lines $CP$ and $XY$ meet at $N$. Find the square of the distance from $X$ to $MN$.
Michael Kural
2014 OMO Fall p29
Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and circumcircle $\Gamma$. It is known that $AB = 7$, $BC = 8$, $CA = 9$. Let $M$ denote the midpoint of major arc $\widehat{BAC}$ of $\Gamma$, and let $D$ denote the intersection of $\Gamma$ with the circumcircle of $\triangle IMO$ (other than $M$). Let $E$ denote the reflection of $D$ over line $IO$. Find the integer closest to $1000 \cdot \frac{BE}{CE}$.
Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and circumcircle $\Gamma$. It is known that $AB = 7$, $BC = 8$, $CA = 9$. Let $M$ denote the midpoint of major arc $\widehat{BAC}$ of $\Gamma$, and let $D$ denote the intersection of $\Gamma$ with the circumcircle of $\triangle IMO$ (other than $M$). Let $E$ denote the reflection of $D$ over line $IO$. Find the integer closest to $1000 \cdot \frac{BE}{CE}$.
Evan Chen
Evan Chen
2015 OMO Spring p13
Let $ABC$ be a scalene triangle whose side lengths are positive integers. It is called stable if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?
Let $ABC$ be a scalene triangle whose side lengths are positive integers. It is called stable if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?
Evan Chen
2015 OMO Spring p14
Let $ABCD$ be a square with side length $2015$. A disk with unit radius is packed neatly inside corner $A$ (i.e. tangent to both $\overline{AB}$ and $\overline{AD}$). Alice kicks the disk, which bounces off $\overline{CD}$, $\overline{BC}$, $\overline{AB}$, $\overline{DA}$, $\overline{DC}$ in that order, before landing neatly into corner $B$. What is the total distance the center of the disk travelled?
Let $ABCD$ be a square with side length $2015$. A disk with unit radius is packed neatly inside corner $A$ (i.e. tangent to both $\overline{AB}$ and $\overline{AD}$). Alice kicks the disk, which bounces off $\overline{CD}$, $\overline{BC}$, $\overline{AB}$, $\overline{DA}$, $\overline{DC}$ in that order, before landing neatly into corner $B$. What is the total distance the center of the disk travelled?
Evan Chen
2015 OMO Spring p17
Let $A,B,M,C,D$ be distinct points on a line such that $AB=BM=MC=CD=6.$ Circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ and radius $4$ and $9$ are tangent to line $AD$ at $A$ and $D$ respectively such that $O_1,O_2$ lie on the same side of line $AD.$ Let $P$ be the point such that $PB\perp O_1M$ and $PC\perp O_2M.$ Determine the value of $PO_2^2-PO_1^2.$
Let $A,B,M,C,D$ be distinct points on a line such that $AB=BM=MC=CD=6.$ Circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ and radius $4$ and $9$ are tangent to line $AD$ at $A$ and $D$ respectively such that $O_1,O_2$ lie on the same side of line $AD.$ Let $P$ be the point such that $PB\perp O_1M$ and $PC\perp O_2M.$ Determine the value of $PO_2^2-PO_1^2.$
Ray Li
2015 OMO Spring p19
Let $ABC$ be a triangle with $AB = 80, BC = 100, AC = 60$. Let $D, E, F$ lie on $BC, AC, AB$ such that $CD = 10, AE = 45, BF = 60$. Let $P$ be a point in the plane of triangle $ABC$. The minimum possible value of $AP+BP+CP+DP+EP+FP$ can be expressed in the form $\sqrt{x}+\sqrt{y}+\sqrt{z}$ for integers $x, y, z$. Find $x+y+z$.
Let $ABC$ be a triangle with $AB = 80, BC = 100, AC = 60$. Let $D, E, F$ lie on $BC, AC, AB$ such that $CD = 10, AE = 45, BF = 60$. Let $P$ be a point in the plane of triangle $ABC$. The minimum possible value of $AP+BP+CP+DP+EP+FP$ can be expressed in the form $\sqrt{x}+\sqrt{y}+\sqrt{z}$ for integers $x, y, z$. Find $x+y+z$.
Yang Liu
2015 OMO Spring p21
Let $A_1A_2A_3A_4A_5$ be a regular pentagon inscribed in a circle with area $\tfrac{5+\sqrt{5}}{10}\pi$. For each $i=1,2,\dots,5$, points $B_i$ and $C_i$ lie on ray $\overrightarrow{A_iA_{i+1}}$ such that
\[B_iA_i \cdot B_iA_{i+1} = B_iA_{i+2} \quad \text{and} \quad C_iA_i \cdot C_iA_{i+1} = C_iA_{i+2}^2\]where indices are taken modulo 5. The value of $\tfrac{[B_1B_2B_3B_4B_5]}{[C_1C_2C_3C_4C_5]}$ (where $[\mathcal P]$ denotes the area of polygon $\mathcal P$) can be expressed as $\tfrac{a+b\sqrt{5}}{c}$, where $a$, $b$, and $c$ are integers, and $c > 0$ is as small as possible. Find $100a+10b+c$.
Let $A_1A_2A_3A_4A_5$ be a regular pentagon inscribed in a circle with area $\tfrac{5+\sqrt{5}}{10}\pi$. For each $i=1,2,\dots,5$, points $B_i$ and $C_i$ lie on ray $\overrightarrow{A_iA_{i+1}}$ such that
\[B_iA_i \cdot B_iA_{i+1} = B_iA_{i+2} \quad \text{and} \quad C_iA_i \cdot C_iA_{i+1} = C_iA_{i+2}^2\]where indices are taken modulo 5. The value of $\tfrac{[B_1B_2B_3B_4B_5]}{[C_1C_2C_3C_4C_5]}$ (where $[\mathcal P]$ denotes the area of polygon $\mathcal P$) can be expressed as $\tfrac{a+b\sqrt{5}}{c}$, where $a$, $b$, and $c$ are integers, and $c > 0$ is as small as possible. Find $100a+10b+c$.
Robin Park
2015 OMO Spring p27
Let $ABCD$ be a quadrilateral satisfying $\angle BCD=\angle CDA$. Suppose rays $AD$ and $BC$ meet at $E$, and let $\Gamma$ be the circumcircle of $ABE$. Let $\Gamma_1$ be a circle tangent to ray $CD$ past $D$ at $W$, segment $AD$ at $X$, and internally tangent to $\Gamma$. Similarly, let $\Gamma_2$ be a circle tangent to ray $DC$ past $C$ at $Y$, segment $BC$ at $Z$, and internally tangent to $\Gamma$. Let $P$ be the intersection of $WX$ and $YZ$, and suppose $P$ lies on $\Gamma$. If $F$ is the $E$-excenter of triangle $ABE$, and $AB=544$, $AE=2197$, $BE=2299$, then find $m+n$, where $FP=\tfrac{m}{n}$ with $m,n$ relatively prime positive integers.
Let $ABCD$ be a quadrilateral satisfying $\angle BCD=\angle CDA$. Suppose rays $AD$ and $BC$ meet at $E$, and let $\Gamma$ be the circumcircle of $ABE$. Let $\Gamma_1$ be a circle tangent to ray $CD$ past $D$ at $W$, segment $AD$ at $X$, and internally tangent to $\Gamma$. Similarly, let $\Gamma_2$ be a circle tangent to ray $DC$ past $C$ at $Y$, segment $BC$ at $Z$, and internally tangent to $\Gamma$. Let $P$ be the intersection of $WX$ and $YZ$, and suppose $P$ lies on $\Gamma$. If $F$ is the $E$-excenter of triangle $ABE$, and $AB=544$, $AE=2197$, $BE=2299$, then find $m+n$, where $FP=\tfrac{m}{n}$ with $m,n$ relatively prime positive integers.
Michael Kural
2015 OMO Spring p29
Let $ABC$ be an acute scalene triangle with incenter $I$, and let $M$ be the circumcenter of triangle $BIC$. Points $D$, $B'$, and $C'$ lie on side $BC$ so that $ \angle BIB' = \angle CIC' = \angle IDB = \angle IDC = 90^{\circ} $. Define $P = \overline{AB} \cap \overline{MC'}$, $Q = \overline{AC} \cap \overline{MB'}$, $S = \overline{MD} \cap \overline{PQ}$, and $K = \overline{SI} \cap \overline{DF}$, where segment $EF$ is a diameter of the incircle selected so that $S$ lies in the interior of segment $AE$. It is known that $KI=15x$, $SI=20x+15$, $BC=20x^{5/2}$, and $DI=20x^{3/2}$, where $x = \tfrac ab(n+\sqrt p)$ for some positive integers $a$, $b$, $n$, $p$, with $p$ prime and $\gcd(a,b)=1$. Compute $a+b+n+p$.
Let $ABC$ be an acute scalene triangle with incenter $I$, and let $M$ be the circumcenter of triangle $BIC$. Points $D$, $B'$, and $C'$ lie on side $BC$ so that $ \angle BIB' = \angle CIC' = \angle IDB = \angle IDC = 90^{\circ} $. Define $P = \overline{AB} \cap \overline{MC'}$, $Q = \overline{AC} \cap \overline{MB'}$, $S = \overline{MD} \cap \overline{PQ}$, and $K = \overline{SI} \cap \overline{DF}$, where segment $EF$ is a diameter of the incircle selected so that $S$ lies in the interior of segment $AE$. It is known that $KI=15x$, $SI=20x+15$, $BC=20x^{5/2}$, and $DI=20x^{3/2}$, where $x = \tfrac ab(n+\sqrt p)$ for some positive integers $a$, $b$, $n$, $p$, with $p$ prime and $\gcd(a,b)=1$. Compute $a+b+n+p$.
Evan Chen
2015 OMO Fall p4
Let $\omega$ be a circle with diameter $AB$ and center $O$. We draw a circle $\omega_A$ through $O$ and $A$, and another circle $\omega_B$ through $O$ and $B$; the circles $\omega_A$ and $\omega_B$ intersect at a point $C$ distinct from $O$. Assume that all three circles $\omega$, $\omega_A$, $\omega_B$ are congruent. If $CO = \sqrt 3$, what is the perimeter of $\triangle ABC$?
Let $\omega$ be a circle with diameter $AB$ and center $O$. We draw a circle $\omega_A$ through $O$ and $A$, and another circle $\omega_B$ through $O$ and $B$; the circles $\omega_A$ and $\omega_B$ intersect at a point $C$ distinct from $O$. Assume that all three circles $\omega$, $\omega_A$, $\omega_B$ are congruent. If $CO = \sqrt 3$, what is the perimeter of $\triangle ABC$?
Evan Chen
2015 OMO Fall p5
Merlin wants to buy a magical box, which happens to be an $n$-dimensional hypercube with side length $1$ cm. The box needs to be large enough to fit his wand, which is $25.6$ cm long.
What is the minimal possible value of $n$?
Merlin wants to buy a magical box, which happens to be an $n$-dimensional hypercube with side length $1$ cm. The box needs to be large enough to fit his wand, which is $25.6$ cm long.
What is the minimal possible value of $n$?
Evan Chen
2015 OMO Fall p6
Farmer John has a (flexible) fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he call enclose an maximum possible area of $A_0$, and when he uses one of the walls or both walls, he gets a maximum of area of $A_1$ and $A_2$ respectively. If $n=\frac{A_1}{A_0}+\frac{A_2}{A_1}$, find $\lfloor 1000n\rfloor$.
Farmer John has a (flexible) fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he call enclose an maximum possible area of $A_0$, and when he uses one of the walls or both walls, he gets a maximum of area of $A_1$ and $A_2$ respectively. If $n=\frac{A_1}{A_0}+\frac{A_2}{A_1}$, find $\lfloor 1000n\rfloor$.
Yannick Yao
2015 OMO Fall p11
A trapezoid $ABCD$ lies on the $xy$-plane. The slopes of lines $BC$ and $AD$ are both $\frac 13$, and the slope of line $AB$ is $-\frac 23$. Given that $AB=CD$ and $BC< AD$, the absolute value of the slope of line $CD$ can be expressed as $\frac mn$, where $m,n$ are two relatively prime positive integers. Find $100m+n$.
A trapezoid $ABCD$ lies on the $xy$-plane. The slopes of lines $BC$ and $AD$ are both $\frac 13$, and the slope of line $AB$ is $-\frac 23$. Given that $AB=CD$ and $BC< AD$, the absolute value of the slope of line $CD$ can be expressed as $\frac mn$, where $m,n$ are two relatively prime positive integers. Find $100m+n$.
Yannick Yao
2015 OMO Fall p15
A regular $2015$-simplex $\mathcal P$ has $2016$ vertices in $2015$-dimensional space such that the distances between every pair of vertices are equal. Let $S$ be the set of points contained inside $\mathcal P$ that are closer to its center than any of its vertices. The ratio of the volume of $S$ to the volume of $\mathcal P$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m+n$ is divided by $1000$.
A regular $2015$-simplex $\mathcal P$ has $2016$ vertices in $2015$-dimensional space such that the distances between every pair of vertices are equal. Let $S$ be the set of points contained inside $\mathcal P$ that are closer to its center than any of its vertices. The ratio of the volume of $S$ to the volume of $\mathcal P$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m+n$ is divided by $1000$.
James Lin
2015 OMO Fall p16
Given a (nondegenrate) triangle $ABC$ with positive integer angles (in degrees), construct squares $BCD_1D_2, ACE_1E_2$ outside the triangle. Given that $D_1, D_2, E_1, E_2$ all lie on a circle, how many ordered triples $(\angle A, \angle B, \angle C)$ are possible?
Given a (nondegenrate) triangle $ABC$ with positive integer angles (in degrees), construct squares $BCD_1D_2, ACE_1E_2$ outside the triangle. Given that $D_1, D_2, E_1, E_2$ all lie on a circle, how many ordered triples $(\angle A, \angle B, \angle C)$ are possible?
Yang Liu
2015 OMO Fall p24
Define $\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2$ for every two points $A = (x_A, y_A)$ and $B = (x_B, y_B)$ in the plane.
Let $S$ be the set of points $(x,y)$ in the plane for which $x,y \in \left\{ 0,1,\dots,100 \right\}$.
Find the number of functions $f : S \to S$ such that $\left\lVert A-B \right\rVert \equiv \left\lVert f(A)-f(B) \right\rVert \pmod{101}$ for any $A, B \in S$.
Define $\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2$ for every two points $A = (x_A, y_A)$ and $B = (x_B, y_B)$ in the plane.
Let $S$ be the set of points $(x,y)$ in the plane for which $x,y \in \left\{ 0,1,\dots,100 \right\}$.
Find the number of functions $f : S \to S$ such that $\left\lVert A-B \right\rVert \equiv \left\lVert f(A)-f(B) \right\rVert \pmod{101}$ for any $A, B \in S$.
Victor Wang
2015 OMO Fall p26
Let $ABC$ be a triangle with $AB=72,AC=98,BC=110$, and circumcircle $\Gamma$, and let $M$ be the midpoint of arc $BC$ not containing $A$ on $\Gamma$. Let $A'$ be the reflection of $A$ over $BC$, and suppose $MB$ meets $AC$ at $D$, while $MC$ meets $AB$ at $E$. If $MA'$ meets $DE$ at $F$, find the distance from $F$ to the center of $\Gamma$.
Let $ABC$ be a triangle with $AB=72,AC=98,BC=110$, and circumcircle $\Gamma$, and let $M$ be the midpoint of arc $BC$ not containing $A$ on $\Gamma$. Let $A'$ be the reflection of $A$ over $BC$, and suppose $MB$ meets $AC$ at $D$, while $MC$ meets $AB$ at $E$. If $MA'$ meets $DE$ at $F$, find the distance from $F$ to the center of $\Gamma$.
Michael Kural
2016 OMO Spring p5
Let $\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possible area of a triangle that is bounded by the $x$-axis, $y$-axis, and $\ell$?
Let $\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possible area of a triangle that is bounded by the $x$-axis, $y$-axis, and $\ell$?
James Lin
Let $ABCDEF$ be a regular hexagon of side length $3$. Let $X, Y,$ and $Z$ be points on segments $AB, CD,$ and $EF$ such that $AX=CY=EZ=1$. The area of triangle $XYZ$ can be expressed in the form $\dfrac{a\sqrt b}{c}$ where $a,b,c$ are positive integers such that $b$ is not divisible by the square of any prime and $\gcd(a,c)=1$. Find $100a+10b+c$.
James Lin
2016 OMO Spring p14
Let $ABC$ be a triangle with $BC=20$ and $CA=16$, and let $I$ be its incenter. If the altitude from $A$ to $BC$, the perpendicular bisector of $AC$, and the line through $I$ perpendicular to $AB$ intersect at a common point, then the length $AB$ can be written as $m+\sqrt{n}$ for positive integers $m$ and $n$. What is $100m+n$?
Let $ABC$ be a triangle with $BC=20$ and $CA=16$, and let $I$ be its incenter. If the altitude from $A$ to $BC$, the perpendicular bisector of $AC$, and the line through $I$ perpendicular to $AB$ intersect at a common point, then the length $AB$ can be written as $m+\sqrt{n}$ for positive integers $m$ and $n$. What is $100m+n$?
Tristan Shin
2016 OMO Spring p22
Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and circumcircle $\omega$. Let $P$ be a point inside $ABC$ such that $PA:PB:PC=2:3:6$. Let rays $\overrightarrow{AP}$, $\overrightarrow{BP}$, and $\overrightarrow{CP}$ intersect $\omega$ again at $X$, $Y$, and $Z$, respectively. The area of $XYZ$ can be expressed in the form $\dfrac{p\sqrt q}{r}$ where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. What is $p+q+r$?
Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and circumcircle $\omega$. Let $P$ be a point inside $ABC$ such that $PA:PB:PC=2:3:6$. Let rays $\overrightarrow{AP}$, $\overrightarrow{BP}$, and $\overrightarrow{CP}$ intersect $\omega$ again at $X$, $Y$, and $Z$, respectively. The area of $XYZ$ can be expressed in the form $\dfrac{p\sqrt q}{r}$ where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. What is $p+q+r$?
James Lin
2016 OMO Spring p27
Let $ABC$ be a triangle with circumradius $2$ and $\angle B-\angle C=15^\circ$. Denote its circumcenter as $O$, orthocenter as $H$, and centroid as $G$. Let the reflection of $H$ over $O$ be $L$, and let lines $AG$ and $AL$ intersect the circumcircle again at $X$ and $Y$, respectively. Define $B_1$ and $C_1$ as the points on the circumcircle of $ABC$ such that $BB_1\parallel AC$ and $CC_1\parallel AB$, and let lines $XY$ and $B_1C_1$ intersect at $Z$. Given that $OZ=2\sqrt 5$, then $AZ^2$ can be expressed in the form $m-\sqrt n$ for positive integers $m$ and $n$. Find $100m+n$.
Let $ABC$ be a triangle with circumradius $2$ and $\angle B-\angle C=15^\circ$. Denote its circumcenter as $O$, orthocenter as $H$, and centroid as $G$. Let the reflection of $H$ over $O$ be $L$, and let lines $AG$ and $AL$ intersect the circumcircle again at $X$ and $Y$, respectively. Define $B_1$ and $C_1$ as the points on the circumcircle of $ABC$ such that $BB_1\parallel AC$ and $CC_1\parallel AB$, and let lines $XY$ and $B_1C_1$ intersect at $Z$. Given that $OZ=2\sqrt 5$, then $AZ^2$ can be expressed in the form $m-\sqrt n$ for positive integers $m$ and $n$. Find $100m+n$.
Michael Ren
2016 OMO Spring p30
In triangle $ABC$, $AB=3\sqrt{30}-\sqrt{10}$, $BC=12$, and $CA=3\sqrt{30}+\sqrt{10}$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AC$. Denote $l$ as the line passing through the circumcenter $O$ and orthocenter $H$ of $ABC$, and let $E$ and $F$ be the feet of the perpendiculars from $B$ and $C$ to $l$, respectively. Let $l'$ be the reflection of $l$ in $BC$ such that $l'$ intersects lines $AE$ and $AF$ at $P$ and $Q$, respectively. Let lines $BP$ and $CQ$ intersect at $K$. $X$, $Y$, and $Z$ are the reflections of $K$ over the perpendicular bisectors of sides $BC$, $CA$, and $AB$, respectively, and $R$ and $S$ are the midpoints of $XY$ and $XZ$, respectively. If lines $MR$ and $NS$ intersect at $T$, then the length of $OT$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $100p+q$.
In triangle $ABC$, $AB=3\sqrt{30}-\sqrt{10}$, $BC=12$, and $CA=3\sqrt{30}+\sqrt{10}$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AC$. Denote $l$ as the line passing through the circumcenter $O$ and orthocenter $H$ of $ABC$, and let $E$ and $F$ be the feet of the perpendiculars from $B$ and $C$ to $l$, respectively. Let $l'$ be the reflection of $l$ in $BC$ such that $l'$ intersects lines $AE$ and $AF$ at $P$ and $Q$, respectively. Let lines $BP$ and $CQ$ intersect at $K$. $X$, $Y$, and $Z$ are the reflections of $K$ over the perpendicular bisectors of sides $BC$, $CA$, and $AB$, respectively, and $R$ and $S$ are the midpoints of $XY$ and $XZ$, respectively. If lines $MR$ and $NS$ intersect at $T$, then the length of $OT$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $100p+q$.
Vincent Huang and James Lin
2016 OMO Fall p3
In a rectangle $ABCD$, let $M$ and $N$ be the midpoints of sides $BC$ and $CD$, respectively, such that $AM$ is perpendicular to $MN$. Given that the length of $AN$ is $60$, the area of rectangle $ABCD$ is $m \sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$.
In a rectangle $ABCD$, let $M$ and $N$ be the midpoints of sides $BC$ and $CD$, respectively, such that $AM$ is perpendicular to $MN$. Given that the length of $AN$ is $60$, the area of rectangle $ABCD$ is $m \sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$.
Yannick Yao
2016 OMO Fall p9
In quadrilateral $ABCD$, $AB=7, BC=24, CD=15, DA=20,$ and $AC=25$. Let segments $AC$ and $BD$ intersect at $E$. What is the length of $EC$?
In quadrilateral $ABCD$, $AB=7, BC=24, CD=15, DA=20,$ and $AC=25$. Let segments $AC$ and $BD$ intersect at $E$. What is the length of $EC$?
James Lin
2016 OMO Fall p13
Let $A_1B_1C_1$ be a triangle with $A_1B_1 = 16, B_1C_1 = 14,$ and $C_1A_1 = 10$. Given a positive integer $i$ and a triangle $A_iB_iC_i$ with circumcenter $O_i$, define triangle $A_{i+1}B_{i+1}C_{i+1}$ in the following way:
(a) $A_{i+1}$ is on side $B_iC_i$ such that $C_iA_{i+1}=2B_iA_{i+1}$.
(b) $B_{i+1}\neq C_i$ is the intersection of line $A_iC_i$ with the circumcircle of $O_iA_{i+1}C_i$.
(c) $C_{i+1}\neq B_i$ is the intersection of line $A_iB_i$ with the circumcircle of $O_iA_{i+1}B_i$.
Find\[ \left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2. \]
Note: $[K]$ denotes the area of $K$.
Let $A_1B_1C_1$ be a triangle with $A_1B_1 = 16, B_1C_1 = 14,$ and $C_1A_1 = 10$. Given a positive integer $i$ and a triangle $A_iB_iC_i$ with circumcenter $O_i$, define triangle $A_{i+1}B_{i+1}C_{i+1}$ in the following way:
(a) $A_{i+1}$ is on side $B_iC_i$ such that $C_iA_{i+1}=2B_iA_{i+1}$.
(b) $B_{i+1}\neq C_i$ is the intersection of line $A_iC_i$ with the circumcircle of $O_iA_{i+1}C_i$.
(c) $C_{i+1}\neq B_i$ is the intersection of line $A_iB_i$ with the circumcircle of $O_iA_{i+1}B_i$.
Find\[ \left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2. \]
Note: $[K]$ denotes the area of $K$.
Yang Liu
2016 OMO Fall p22
Let $ABC$ be a triangle with $AB=3$ and $AC=4$. It is given that there does not exist a point $D$, different from $A$ and not lying on line $BC$, such that the Euler line of $ABC$ coincides with the Euler line of $DBC$. The square of the product of all possible lengths of $BC$ can be expressed in the form $m+n\sqrt p$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $100m+10n+p$.
Note: For this problem, consider every line passing through the center of an equilateral triangle to be an Euler line of the equilateral triangle. Hence, if $D$ is chosen such that $DBC$ is an equilateral triangle and the Euler line of $ABC$ passes through the center of $DBC$, then consider the Euler line of $ABC$ to coincide with "the" Euler line of $DBC$.
Let $ABC$ be a triangle with $AB=3$ and $AC=4$. It is given that there does not exist a point $D$, different from $A$ and not lying on line $BC$, such that the Euler line of $ABC$ coincides with the Euler line of $DBC$. The square of the product of all possible lengths of $BC$ can be expressed in the form $m+n\sqrt p$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $100m+10n+p$.
Note: For this problem, consider every line passing through the center of an equilateral triangle to be an Euler line of the equilateral triangle. Hence, if $D$ is chosen such that $DBC$ is an equilateral triangle and the Euler line of $ABC$ passes through the center of $DBC$, then consider the Euler line of $ABC$ to coincide with "the" Euler line of $DBC$.
Michael Ren
2016 OMO Fall p26
Let $ABC$ be a triangle with $BC=9$, $CA=8$, and $AB=10$. Let the incenter and incircle of $ABC$ be $I$ and $\gamma$, respectively, and let $N$ be the midpoint of major arc $BC$ of the cirucmcircle of $ABC$. Line $NI$ meets the circumcircle of $ABC$ a second time at $P$. Let the line through $I$ perpendicular to $AI$ meet segments $AB$, $AC$, and $AP$ at $C_1$, $B_1$, and $Q$, respectively. Let $B_2$ lie on segment $CQ$ such that line $B_1B_2$ is tangent to $\gamma$, and let $C_2$ lie on segment $BQ$ such that line $C_1C_2$ tangent to $\gamma$. The length of $B_2C_2$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $100m+n$.
Let $ABC$ be a triangle with $BC=9$, $CA=8$, and $AB=10$. Let the incenter and incircle of $ABC$ be $I$ and $\gamma$, respectively, and let $N$ be the midpoint of major arc $BC$ of the cirucmcircle of $ABC$. Line $NI$ meets the circumcircle of $ABC$ a second time at $P$. Let the line through $I$ perpendicular to $AI$ meet segments $AB$, $AC$, and $AP$ at $C_1$, $B_1$, and $Q$, respectively. Let $B_2$ lie on segment $CQ$ such that line $B_1B_2$ is tangent to $\gamma$, and let $C_2$ lie on segment $BQ$ such that line $C_1C_2$ tangent to $\gamma$. The length of $B_2C_2$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $100m+n$.
Vincent Huang
2016 OMO Fall p28
Let $ABC$ be a triangle with $AB=34,BC=25,$ and $CA=39$. Let $O,H,$ and $ \omega$ be the circumcenter, orthocenter, and circumcircle of $\triangle ABC$, respectively. Let line $AH$ meet $\omega$ a second time at $A_1$ and let the reflection of $H$ over the perpendicular bisector of $BC$ be $H_1$. Suppose the line through $O$ perpendicular to $A_1O$ meets $\omega$ at two points $Q$ and $R$ with $Q$ on minor arc $AC$ and $R$ on minor arc $AB$. Denote $\mathcal H$ as the hyperbola passing through $A,B,C,H,H_1$, and suppose $HO$ meets $\mathcal H$ again at $P$. Let $X,Y$ be points with $XH \parallel AR \parallel YP, XP \parallel AQ \parallel YH$. Let $P_1,P_2$ be points on the tangent to $\mathcal H$ at $P$ with $XP_1 \parallel OH \parallel YP_2$ and let $P_3,P_4$ be points on the tangent to $\mathcal H$ at $H$ with $XP_3 \parallel OH \parallel YP_4$. If $P_1P_4$ and $P_2P_3$ meet at $N$, and $ON$ may be written in the form $\frac{a}{b}$ where $a,b$ are positive coprime integers, find $100a+b$.
Let $ABC$ be a triangle with $AB=34,BC=25,$ and $CA=39$. Let $O,H,$ and $ \omega$ be the circumcenter, orthocenter, and circumcircle of $\triangle ABC$, respectively. Let line $AH$ meet $\omega$ a second time at $A_1$ and let the reflection of $H$ over the perpendicular bisector of $BC$ be $H_1$. Suppose the line through $O$ perpendicular to $A_1O$ meets $\omega$ at two points $Q$ and $R$ with $Q$ on minor arc $AC$ and $R$ on minor arc $AB$. Denote $\mathcal H$ as the hyperbola passing through $A,B,C,H,H_1$, and suppose $HO$ meets $\mathcal H$ again at $P$. Let $X,Y$ be points with $XH \parallel AR \parallel YP, XP \parallel AQ \parallel YH$. Let $P_1,P_2$ be points on the tangent to $\mathcal H$ at $P$ with $XP_1 \parallel OH \parallel YP_2$ and let $P_3,P_4$ be points on the tangent to $\mathcal H$ at $H$ with $XP_3 \parallel OH \parallel YP_4$. If $P_1P_4$ and $P_2P_3$ meet at $N$, and $ON$ may be written in the form $\frac{a}{b}$ where $a,b$ are positive coprime integers, find $100a+b$.
Vincent Huang
In rectangle $ABCD$, $AB=6$ and $BC=16$. Points $P, Q$ are chosen on the interior of side $AB$ such that $AP=PQ=QB$, and points $R, S$ are chosen on the interior of side $CD$ such that $CR=RS=SD$. Find the area of the region formed by the union of parallelograms $APCR$ and $QBSD$.
Let $ABC$ be a triangle, not right-angled, with positive integer angle measures (in degrees) and circumcenter $O$. Say that a triangle $ABC$ is good if the following three conditions hold:
(a) There exists a point $P\neq A$ on side $AB$ such that the circumcircle of $\triangle POA$ is tangent to $BO$.
(b) There exists a point $Q\neq A$ on side $AC$ such that the circumcircle of $\triangle QOA$ is tangent to $CO$.
(c) The perimeter of $\triangle APQ$ is at least $AB+AC$.
Determine the number of ordered triples $(\angle A, \angle B,\angle C)$ for which $\triangle ABC$ is good.
Yannick Yao
2017 OMO Spring p6
Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$. $X$, $Y$, and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$, $Y$ is on minor arc $CD$, and $Z$ is on minor arc $EF$, where $X$ may coincide with $A$ or $B$ (And similarly for $Y$ and $Z$). Compute the square of the smallest possible area of $XYZ$.
Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$. $X$, $Y$, and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$, $Y$ is on minor arc $CD$, and $Z$ is on minor arc $EF$, where $X$ may coincide with $A$ or $B$ (And similarly for $Y$ and $Z$). Compute the square of the smallest possible area of $XYZ$.
Michael Ren
2017 OMO Spring p14Let $ABC$ be a triangle, not right-angled, with positive integer angle measures (in degrees) and circumcenter $O$. Say that a triangle $ABC$ is good if the following three conditions hold:
(a) There exists a point $P\neq A$ on side $AB$ such that the circumcircle of $\triangle POA$ is tangent to $BO$.
(b) There exists a point $Q\neq A$ on side $AC$ such that the circumcircle of $\triangle QOA$ is tangent to $CO$.
(c) The perimeter of $\triangle APQ$ is at least $AB+AC$.
Determine the number of ordered triples $(\angle A, \angle B,\angle C)$ for which $\triangle ABC$ is good.
Vincent Huang
2017 OMO Spring p17
Let $ABC$ be a triangle with $BC=7,AB=5$, and $AC=8$. Let $M,N$ be the midpoints of sides $AC,AB$ respectively, and let $O$ be the circumcenter of $ABC$. Let $BO, CO$ meet $AC, AB$ at $P$ and $Q$, respectively. If $MN$ meets $PQ$ at $R$ and $OR$ meets $BC$ at $S$, then the value of $OS^2$ can be written in the form $\frac{m}{n}$ where $m,n$ are relatively prime positive integers. Find $100m+n$.
Let $ABC$ be a triangle with $BC=7,AB=5$, and $AC=8$. Let $M,N$ be the midpoints of sides $AC,AB$ respectively, and let $O$ be the circumcenter of $ABC$. Let $BO, CO$ meet $AC, AB$ at $P$ and $Q$, respectively. If $MN$ meets $PQ$ at $R$ and $OR$ meets $BC$ at $S$, then the value of $OS^2$ can be written in the form $\frac{m}{n}$ where $m,n$ are relatively prime positive integers. Find $100m+n$.
Vincent Huang
2017 OMO Spring p26
Let $ABC$ be a triangle with $AB=13,BC=15,AC=14$, circumcenter $O$, and orthocenter $H$, and let $M,N$ be the midpoints of minor and major arcs $BC$ on the circumcircle of $ABC$. Suppose $P\in AB, Q\in AC$ satisfy that $P,O,Q$ are collinear and $PQ||AN$, and point $I$ satisfies $IP\perp AB,IQ\perp AC$. Let $H'$ be the reflection of $H$ over line $PQ$, and suppose $H'I$ meets $PQ$ at a point $T$. If $\frac{MT}{NT}$ can be written in the form $\frac{\sqrt{m}}{n}$ for positive integers $m,n$ where $m$ is not divisible by the square of any prime, then find $100m+n$.
Vincent Huang
2017 OMO Spring p29
Let $ABC$ be a triangle with $AB=2\sqrt6, BC=5, CA=\sqrt{26}$, midpoint $M$ of $BC$, circumcircle $\Omega$, and orthocenter $H$. Let $BH$ intersect $AC$ at $E$ and $CH$ intersect $AB$ at $F$. Let $R$ be the midpoint of $EF$ and let $N$ be the midpoint of $AH$. Let $AR$ intersect the circumcircle of $AHM$ again at $L$. Let the circumcircle of $ANL$ intersect $\Omega$ and the circumcircle of $BNC$ at $J$ and $O$, respectively. Let circles $AHM$ and $JMO$ intersect again at $U$, and let $AU$ intersect the circumcircle of $AHC$ again at $V \neq A$. The square of the length of $CV$ can be expressed in the form $\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
Let $ABC$ be a triangle with $AB=2\sqrt6, BC=5, CA=\sqrt{26}$, midpoint $M$ of $BC$, circumcircle $\Omega$, and orthocenter $H$. Let $BH$ intersect $AC$ at $E$ and $CH$ intersect $AB$ at $F$. Let $R$ be the midpoint of $EF$ and let $N$ be the midpoint of $AH$. Let $AR$ intersect the circumcircle of $AHM$ again at $L$. Let the circumcircle of $ANL$ intersect $\Omega$ and the circumcircle of $BNC$ at $J$ and $O$, respectively. Let circles $AHM$ and $JMO$ intersect again at $U$, and let $AU$ intersect the circumcircle of $AHC$ again at $V \neq A$. The square of the length of $CV$ can be expressed in the form $\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
Michael Ren
2017 OMO Fall p6
A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$.
A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$.
Yannick Yao
Bill draws two circles which intersect at $X,Y$. Let $P$ be the intersection of the common tangents to the two circles and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the radii of the two circles are $3,4$ and the distance between the centers of these two circles is $5$, then the largest distance from $Q$ to any point on either of the circles can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Tristan Shin
2017 OMO Fall p16
Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two parabolas with distinct directrices $\ell_1$ and $\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is known that $F_1F_2||\ell_1||\ell_2$, $F_1$ lies on $\mathcal{P}_2$, and $F_2$ lies on $\mathcal{P}_1$. The two parabolas intersect at distinct points $A$ and $B$. Given that $F_1F_2=1$, the value of $AB^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two parabolas with distinct directrices $\ell_1$ and $\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is known that $F_1F_2||\ell_1||\ell_2$, $F_1$ lies on $\mathcal{P}_2$, and $F_2$ lies on $\mathcal{P}_1$. The two parabolas intersect at distinct points $A$ and $B$. Given that $F_1F_2=1$, the value of $AB^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
Yannick Yao
2017 OMO Fall p28
Let $ABC$ be a triangle with $AB=7, AC=9, BC=10$, circumcenter $O$, circumradius $R$, and circumcircle $\omega$. Let the tangents to $\omega$ at $B,C$ meet at $X$. A variable line $\ell$ passes through $O$. Let $A_1$ be the projection of $X$ onto $\ell$ and $A_2$ be the reflection of $A_1$ over $O$. Suppose that there exist two points $Y,Z$ on $\ell$ such that $\angle YAB+\angle YBC+\angle YCA=\angle ZAB+\angle ZBC+\angle ZCA=90^{\circ}$, where all angles are directed, and furthermore that $O$ lies inside segment $YZ$ with $OY*OZ=R^2$. Then there are several possible values for the sine of the angle at which the angle bisector of $\angle AA_2O$ meets $BC$. If the product of these values can be expressed in the form $\frac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $b$ squarefree and $a,c$ coprime, determine $a+b+c$.
Let $ABC$ be a triangle with $AB=7, AC=9, BC=10$, circumcenter $O$, circumradius $R$, and circumcircle $\omega$. Let the tangents to $\omega$ at $B,C$ meet at $X$. A variable line $\ell$ passes through $O$. Let $A_1$ be the projection of $X$ onto $\ell$ and $A_2$ be the reflection of $A_1$ over $O$. Suppose that there exist two points $Y,Z$ on $\ell$ such that $\angle YAB+\angle YBC+\angle YCA=\angle ZAB+\angle ZBC+\angle ZCA=90^{\circ}$, where all angles are directed, and furthermore that $O$ lies inside segment $YZ$ with $OY*OZ=R^2$. Then there are several possible values for the sine of the angle at which the angle bisector of $\angle AA_2O$ meets $BC$. If the product of these values can be expressed in the form $\frac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $b$ squarefree and $a,c$ coprime, determine $a+b+c$.
Vincent Huang
2017 OMO Fall p30
We define the bulldozer of triangle $ABC$ as the segment between points $P$ and $Q$, distinct points in the plane of $ABC$ such that $PA\cdot BC=PB\cdot CA=PC\cdot AB$ and $QA\cdot BC=QB\cdot CA=QC\cdot AB$. Let $XY$ be a segment of unit length in a plane $\mathcal{P}$, and let $\mathcal{S}$ be the region of $\mathcal P$ that the bulldozer of $XYZ$ sweeps through as $Z$ varies across the points in $\mathcal{P}$ satisfying $XZ=2YZ$. Find the greatest integer that is less than $100$ times the area of $\mathcal S$.
We define the bulldozer of triangle $ABC$ as the segment between points $P$ and $Q$, distinct points in the plane of $ABC$ such that $PA\cdot BC=PB\cdot CA=PC\cdot AB$ and $QA\cdot BC=QB\cdot CA=QC\cdot AB$. Let $XY$ be a segment of unit length in a plane $\mathcal{P}$, and let $\mathcal{S}$ be the region of $\mathcal P$ that the bulldozer of $XYZ$ sweeps through as $Z$ varies across the points in $\mathcal{P}$ satisfying $XZ=2YZ$. Find the greatest integer that is less than $100$ times the area of $\mathcal S$.
Michael Ren
2018 OMO Spring p2
The area of a circle (in square inches) is numerically larger than its circumference (in inches). What is the smallest possible integral area of the circle, in square inches?
The area of a circle (in square inches) is numerically larger than its circumference (in inches). What is the smallest possible integral area of the circle, in square inches?
James Lin
2018 OMO Spring p7
A quadrilateral and a pentagon (both not self-intersecting) intersect each other at $N$ distinct points, where $N$ is a positive integer. What is the maximal possible value of $N$?
A quadrilateral and a pentagon (both not self-intersecting) intersect each other at $N$ distinct points, where $N$ is a positive integer. What is the maximal possible value of $N$?
James Lin
2018 OMO Spring p9
Let $k$ be a positive integer. In the coordinate plane, circle $\omega$ has positive integer radius and is tangent to both axes. Suppose that $\omega$ passes through $(1,1000+k)$. Compute the smallest possible value of $k$.
Let $k$ be a positive integer. In the coordinate plane, circle $\omega$ has positive integer radius and is tangent to both axes. Suppose that $\omega$ passes through $(1,1000+k)$. Compute the smallest possible value of $k$.
Luke Robitaille
2018 OMO Spring p14
Let $ABC$ be a triangle with $AB=20$ and $AC=18$. $E$ is on segment $AC$ and $F$ is on segment $AB$ such that $AE=AF=8$. Let $BE$ and $CF$ intersect at $G$. Given that $AEGF$ is cyclic, then $BC=m\sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$.
Let $ABC$ be a triangle with $AB=20$ and $AC=18$. $E$ is on segment $AC$ and $F$ is on segment $AB$ such that $AE=AF=8$. Let $BE$ and $CF$ intersect at $G$. Given that $AEGF$ is cyclic, then $BC=m\sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$.
James Lin
2018 OMO Spring p20
Let $ABC$ be a triangle with $AB = 7, BC = 5,$ and $CA = 6$. Let $D$ be a variable point on segment $BC$, and let the perpendicular bisector of $AD$ meet segments $AC, AB$ at $E, F,$ respectively. It is given that there is a point $P$ inside $\triangle ABC$ such that $\frac{AP}{PC} = \frac{AE}{EC}$ and $\frac{AP}{PB} = \frac{AF}{FB}$. The length of the path traced by $P$ as $D$ varies along segment $BC$ can be expressed as $\sqrt{\frac{m}{n}}\sin^{-1}\left(\sqrt \frac 17\right)$, where $m$ and $n$ are relatively prime positive integers, and angles are measured in radians. Compute $100m + n$.
Let $ABC$ be a triangle with $AB = 7, BC = 5,$ and $CA = 6$. Let $D$ be a variable point on segment $BC$, and let the perpendicular bisector of $AD$ meet segments $AC, AB$ at $E, F,$ respectively. It is given that there is a point $P$ inside $\triangle ABC$ such that $\frac{AP}{PC} = \frac{AE}{EC}$ and $\frac{AP}{PB} = \frac{AF}{FB}$. The length of the path traced by $P$ as $D$ varies along segment $BC$ can be expressed as $\sqrt{\frac{m}{n}}\sin^{-1}\left(\sqrt \frac 17\right)$, where $m$ and $n$ are relatively prime positive integers, and angles are measured in radians. Compute $100m + n$.
Edward Wan
2018 OMO Spring p23
Let $ABC$ be a triangle with $BC=13, CA=11, AB=10$. Let $A_1$ be the midpoint of $BC$. A variable line $\ell$ passes through $A_1$ and meets $AC,AB$ at $B_1,C_1$. Let $B_2,C_2$ be points with $B_2B=B_2C, B_2C_1\perp AB, C_2B=C_2C, C_2B_1 \perp AC$, and define $P=BB_2\cap CC_2$. Suppose the circles of diameters $BB_2, CC_2$ meet at a point $Q\neq A_1$. Given that $Q$ lies on the same side of line $BC$ as $A$, the minimum possible value of $\dfrac{PB}{PC}+\dfrac{QB}{QC}$ can be expressed in the form $\dfrac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $\gcd (a,c)=1$ and $b$ squarefree. Determine $a+b+c$.
Let $ABC$ be a triangle with $BC=13, CA=11, AB=10$. Let $A_1$ be the midpoint of $BC$. A variable line $\ell$ passes through $A_1$ and meets $AC,AB$ at $B_1,C_1$. Let $B_2,C_2$ be points with $B_2B=B_2C, B_2C_1\perp AB, C_2B=C_2C, C_2B_1 \perp AC$, and define $P=BB_2\cap CC_2$. Suppose the circles of diameters $BB_2, CC_2$ meet at a point $Q\neq A_1$. Given that $Q$ lies on the same side of line $BC$ as $A$, the minimum possible value of $\dfrac{PB}{PC}+\dfrac{QB}{QC}$ can be expressed in the form $\dfrac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $\gcd (a,c)=1$ and $b$ squarefree. Determine $a+b+c$.
Vincent Huang
2018 OMO Spring p26
Let $ABC$ be a triangle with incenter $I$. Let $P$ and $Q$ be points such that $IP\perp AC$, $IQ\perp AB$, and $IA\perp PQ$. Assume that $BP$ and $CQ$ intersect at the point $R\neq A$ on the circumcircle of $ABC$ such that $AR\parallel BC$. Given that $\angle B-\angle C=36^\circ$, the value of $\cos A$ can be expressed in the form $\frac{m-\sqrt n}{p}$ for positive integers $m,n,p$ and where $n$ is not divisible by the square of any prime. Find the value of $100m+10n+p$.
Let $ABC$ be a triangle with incenter $I$. Let $P$ and $Q$ be points such that $IP\perp AC$, $IQ\perp AB$, and $IA\perp PQ$. Assume that $BP$ and $CQ$ intersect at the point $R\neq A$ on the circumcircle of $ABC$ such that $AR\parallel BC$. Given that $\angle B-\angle C=36^\circ$, the value of $\cos A$ can be expressed in the form $\frac{m-\sqrt n}{p}$ for positive integers $m,n,p$ and where $n$ is not divisible by the square of any prime. Find the value of $100m+10n+p$.
Michael Ren
2018 OMO Spring p28
In $\triangle ABC$, the incircle $\omega$ has center $I$ and is tangent to $\overline{CA}$ and $\overline{AB}$ at $E$ and $F$ respectively. The circumcircle of $\triangle{BIC}$ meets $\omega$ at $P$ and $Q$. Lines $AI$ and $BC$ meet at $D$, and the circumcircle of $\triangle PDQ$ meets $\overline{BC}$ again at $X$. Suppose that $EF = PQ = 16$ and $PX + QX = 17$. Then $BC^2$ can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$.
In $\triangle ABC$, the incircle $\omega$ has center $I$ and is tangent to $\overline{CA}$ and $\overline{AB}$ at $E$ and $F$ respectively. The circumcircle of $\triangle{BIC}$ meets $\omega$ at $P$ and $Q$. Lines $AI$ and $BC$ meet at $D$, and the circumcircle of $\triangle PDQ$ meets $\overline{BC}$ again at $X$. Suppose that $EF = PQ = 16$ and $PX + QX = 17$. Then $BC^2$ can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$.
Ankan Bhattacharya and Michael Ren
2018 OMO Fall p5
In triangle $ABC$, $AB=8, AC=9,$ and $BC=10$. Let $M$ be the midpoint of $BC$. Circle $\omega_1$ with area $A_1$ passes through $A,B,$ and $C$. Circle $\omega_2$ with area $A_2$ passes through $A,B,$ and $M$. Then $\frac{A_1}{A_2}=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
In triangle $ABC$, $AB=8, AC=9,$ and $BC=10$. Let $M$ be the midpoint of $BC$. Circle $\omega_1$ with area $A_1$ passes through $A,B,$ and $C$. Circle $\omega_2$ with area $A_2$ passes through $A,B,$ and $M$. Then $\frac{A_1}{A_2}=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Luke Robitaille
2018 OMO Fall p8
Let $ABC$ be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let $N = 2017$, and define the $A$-ntipodes to be the points $A_1,\dots, A_N$ to be the points on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$, and similarly define the $B$, $C$-ntipodes. A line $\ell_A$ through $A$ is called a qevian if it passes through an $A$-ntipode, and similarly we define qevians through $B$ and $C$. Compute the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$, $B$, $C$, respectively.
Let $ABC$ be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let $N = 2017$, and define the $A$-ntipodes to be the points $A_1,\dots, A_N$ to be the points on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$, and similarly define the $B$, $C$-ntipodes. A line $\ell_A$ through $A$ is called a qevian if it passes through an $A$-ntipode, and similarly we define qevians through $B$ and $C$. Compute the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$, $B$, $C$, respectively.
Brandon Wang
2018 OMO Fall p14
In triangle $ABC$, $AB=13, BC=14, CA=15$. Let $\Omega$ and $\omega$ be the circumcircle and incircle of $ABC$ respectively. Among all circles that are tangent to both $\Omega$ and $\omega$, call those that contain $\omega$ inclusive and those that do not contain $\omega$ exclusive. Let $\mathcal{I}$ and $\mathcal{E}$ denote the set of centers of inclusive circles and exclusive circles respectively, and let $I$ and $E$ be the area of the regions enclosed by $\mathcal{I}$ and $\mathcal{E}$ respectively. The ratio $\frac{I}{E}$ can be expressed as $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
In triangle $ABC$, $AB=13, BC=14, CA=15$. Let $\Omega$ and $\omega$ be the circumcircle and incircle of $ABC$ respectively. Among all circles that are tangent to both $\Omega$ and $\omega$, call those that contain $\omega$ inclusive and those that do not contain $\omega$ exclusive. Let $\mathcal{I}$ and $\mathcal{E}$ denote the set of centers of inclusive circles and exclusive circles respectively, and let $I$ and $E$ be the area of the regions enclosed by $\mathcal{I}$ and $\mathcal{E}$ respectively. The ratio $\frac{I}{E}$ can be expressed as $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Yannick Yao
2018 OMO Fall p17
A hyperbola in the coordinate plane passing through the points $(2,5)$, $(7,3)$, $(1,1)$, and $(10,10)$ has an asymptote of slope $\frac{20}{17}$. The slope of its other asymptote can be expressed in the form $-\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
A hyperbola in the coordinate plane passing through the points $(2,5)$, $(7,3)$, $(1,1)$, and $(10,10)$ has an asymptote of slope $\frac{20}{17}$. The slope of its other asymptote can be expressed in the form $-\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Michael Ren
Let $ABC$ be a triangle with $AB=2$ and $AC=3$. Let $H$ be the orthocenter, and let $M$ be the midpoint of $BC$. Let the line through $H$ perpendicular to line $AM$ intersect line $AB$ at $X$ and line $AC$ at $Y$. Suppose that lines $BY$ and $CX$ are parallel. Then $[ABC]^2=\frac{a+b\sqrt{c}}{d}$ for positive integers $a,b,c$ and $d$, where $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime. Compute $1000a+100b+10c+d$.
Luke Robitaille
2018 OMO Fall p28
Let $\omega$ be a circle centered at $O$ with radius $R=2018$. For any $0 < r < 1009$, let $\gamma$ be a circle of radius $r$ centered at a point $I$ satisfying $OI =\sqrt{R(R-2r)}$. Choose any $A,B,C\in \omega$ with $AC, AB$ tangent to $\gamma$ at $E,F$, respectively. Suppose a circle of radius $r_A$ is tangent to $AB,AC$, and internally tangent to $\omega$ at a point $D$ with $r_A=5r$. Let line $EF$ meet $\omega$ at $P_1,Q_1$. Suppose $P_2,P_3,Q_2,Q_3$ lie on $\omega$ such that $P_1P_2,P_1P_3,Q_1Q_2,Q_1Q_3$ are tangent to $\gamma$. Let $P_2P_3,Q_2Q_3$ meet at $K$, and suppose $KI$ meets $AD$ at a point $X$. Then as $r$ varies from $0$ to $1009$, the maximum possible value of $OX$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers such that $b$ is not divisible by the square of any prime and $\gcd (a,c)=1$. Compute $10a+b+c$.
Let $\omega$ be a circle centered at $O$ with radius $R=2018$. For any $0 < r < 1009$, let $\gamma$ be a circle of radius $r$ centered at a point $I$ satisfying $OI =\sqrt{R(R-2r)}$. Choose any $A,B,C\in \omega$ with $AC, AB$ tangent to $\gamma$ at $E,F$, respectively. Suppose a circle of radius $r_A$ is tangent to $AB,AC$, and internally tangent to $\omega$ at a point $D$ with $r_A=5r$. Let line $EF$ meet $\omega$ at $P_1,Q_1$. Suppose $P_2,P_3,Q_2,Q_3$ lie on $\omega$ such that $P_1P_2,P_1P_3,Q_1Q_2,Q_1Q_3$ are tangent to $\gamma$. Let $P_2P_3,Q_2Q_3$ meet at $K$, and suppose $KI$ meets $AD$ at a point $X$. Then as $r$ varies from $0$ to $1009$, the maximum possible value of $OX$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers such that $b$ is not divisible by the square of any prime and $\gcd (a,c)=1$. Compute $10a+b+c$.
Vincent Huang
2018 OMO Fall p30
Let $ABC$ be an acute triangle with $\cos B =\frac{1}{3}, \cos C =\frac{1}{4}$, and circumradius $72$. Let $ABC$ have circumcenter $O$, symmedian point $K$, and nine-point center $N$. Consider all non-degenerate hyperbolas $\mathcal H$ with perpendicular asymptotes passing through $A,B,C$. Of these $\mathcal H$, exactly one has the property that there exists a point $P\in \mathcal H$ such that $NP$ is tangent to $\mathcal H$ and $P\in OK$. Let $N'$ be the reflection of $N$ over $BC$. If $AK$ meets $PN'$ at $Q$, then the length of $PQ$ can be expressed in the form $a+b\sqrt{c}$, where $a,b,c$ are positive integers such that $c$ is not divisible by the square of any prime. Compute $100a+b+c$.
Let $ABC$ be an acute triangle with $\cos B =\frac{1}{3}, \cos C =\frac{1}{4}$, and circumradius $72$. Let $ABC$ have circumcenter $O$, symmedian point $K$, and nine-point center $N$. Consider all non-degenerate hyperbolas $\mathcal H$ with perpendicular asymptotes passing through $A,B,C$. Of these $\mathcal H$, exactly one has the property that there exists a point $P\in \mathcal H$ such that $NP$ is tangent to $\mathcal H$ and $P\in OK$. Let $N'$ be the reflection of $N$ over $BC$. If $AK$ meets $PN'$ at $Q$, then the length of $PQ$ can be expressed in the form $a+b\sqrt{c}$, where $a,b,c$ are positive integers such that $c$ is not divisible by the square of any prime. Compute $100a+b+c$.
Vincent Huang
2019 OMO Spring p7
Let $ABCD$ be a square with side length $4$. Consider points $P$ and $Q$ on segments $AB$ and $BC$, respectively, with $BP=3$ and $BQ=1$. Let $R$ be the intersection of $AQ$ and $DP$. If $BR^2$ can be expressed in the form $\frac{m}{n}$ for coprime positive integers $m,n$, compute $m+n$.
Let $ABCD$ be a square with side length $4$. Consider points $P$ and $Q$ on segments $AB$ and $BC$, respectively, with $BP=3$ and $BQ=1$. Let $R$ be the intersection of $AQ$ and $DP$. If $BR^2$ can be expressed in the form $\frac{m}{n}$ for coprime positive integers $m,n$, compute $m+n$.
Brandon Wang
2019 OMO Spring p8
In triangle $ABC$, side $AB$ has length $10$, and the $A$- and $B$-medians have length $9$ and $12$, respectively. Compute the area of the triangle.
In triangle $ABC$, side $AB$ has length $10$, and the $A$- and $B$-medians have length $9$ and $12$, respectively. Compute the area of the triangle.
Yannick Yao
2019 OMO Spring p16
In triangle $ABC$, $BC=3, CA=4$, and $AB=5$. For any point $P$ in the same plane as $ABC$, define $f(P)$ as the sum of the distances from $P$ to lines $AB, BC$, and $CA$. The area of the locus of $P$ where $f(P)\leq 12$ is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
In triangle $ABC$, $BC=3, CA=4$, and $AB=5$. For any point $P$ in the same plane as $ABC$, define $f(P)$ as the sum of the distances from $P$ to lines $AB, BC$, and $CA$. The area of the locus of $P$ where $f(P)\leq 12$ is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Yannick Yao
2019 OMO Spring p17
Let $ABCD$ be an isosceles trapezoid with $\overline{AD} \parallel \overline{BC}$. The incircle of $\triangle ABC$ has center $I$ and is tangent to $\overline{BC}$ at $P$. The incircle of $\triangle ABD$ has center $J$ and is tangent to $\overline{AD}$ at $Q$. If $PI = 8$, $IJ = 25$, and $JQ = 15$, compute the greatest integer less than or equal to the area of $ABCD$.
Let $ABCD$ be an isosceles trapezoid with $\overline{AD} \parallel \overline{BC}$. The incircle of $\triangle ABC$ has center $I$ and is tangent to $\overline{BC}$ at $P$. The incircle of $\triangle ABD$ has center $J$ and is tangent to $\overline{AD}$ at $Q$. If $PI = 8$, $IJ = 25$, and $JQ = 15$, compute the greatest integer less than or equal to the area of $ABCD$.
Ankan Bhattacharya
2019 OMO Spring p20
Let $ABC$ be a triangle with $AB=4$, $BC=5$, and $CA=6$. Suppose $X$ and $Y$ are points such that $BC$ and $XY$ are parallel, $BX$ and $CY$ intersect at a point $P$ on the circumcircle of $\triangle{ABC}$, the circumcircles of $\triangle{BCX}$ and $\triangle{BCY}$ are tangent to $AB$ and $AC$, respectively.
Then $AP^2$ can be written in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Compute $100p+q$.
Let $ABC$ be a triangle with $AB=4$, $BC=5$, and $CA=6$. Suppose $X$ and $Y$ are points such that $BC$ and $XY$ are parallel, $BX$ and $CY$ intersect at a point $P$ on the circumcircle of $\triangle{ABC}$, the circumcircles of $\triangle{BCX}$ and $\triangle{BCY}$ are tangent to $AB$ and $AC$, respectively.
Then $AP^2$ can be written in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Compute $100p+q$.
Tristan Shin
2019 OMO Spring p28
Let $ABC$ be a triangle. There exists a positive real number $x$ such that $AB=6x^2+1$ and $AC = 2x^2+2x$, and there exist points $W$ and $X$ on segment $AB$ along with points $Y$ and $Z$ on segment $AC$ such that $AW=x$, $WX=x+4$, $AY=x+1$, and $YZ=x$. For any line $\ell$ not intersecting segment $BC$, let $f(\ell)$ be the unique point $P$ on line $\ell$ and on the same side of $BC$ as $A$ such that $\ell$ is tangent to the circumcircle of triangle $PBC$. Suppose lines $f(WY)f(XY)$ and $f(WZ)f(XZ)$ meet at $B$, and that lines $f(WZ)f(WY)$ and $f(XY)f(XZ)$ meet at $C$. Then the product of all possible values for the length of $BC$ can be expressed in the form $a + \dfrac{b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $c$ squarefree and $\gcd (b,d)=1$. Compute $100a+b+c+d$.
Let $ABC$ be a triangle. There exists a positive real number $x$ such that $AB=6x^2+1$ and $AC = 2x^2+2x$, and there exist points $W$ and $X$ on segment $AB$ along with points $Y$ and $Z$ on segment $AC$ such that $AW=x$, $WX=x+4$, $AY=x+1$, and $YZ=x$. For any line $\ell$ not intersecting segment $BC$, let $f(\ell)$ be the unique point $P$ on line $\ell$ and on the same side of $BC$ as $A$ such that $\ell$ is tangent to the circumcircle of triangle $PBC$. Suppose lines $f(WY)f(XY)$ and $f(WZ)f(XZ)$ meet at $B$, and that lines $f(WZ)f(WY)$ and $f(XY)f(XZ)$ meet at $C$. Then the product of all possible values for the length of $BC$ can be expressed in the form $a + \dfrac{b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $c$ squarefree and $\gcd (b,d)=1$. Compute $100a+b+c+d$.
Vincent Huang
2019 OMO Spring p30
Let $ABC$ be a triangle with symmedian point $K$, and let $\theta = \angle AKB-90^{\circ}$. Suppose that $\theta$ is both positive and less than $\angle C$. Consider a point $K'$ inside $\triangle ABC$ such that $A,K',K,$ and $B$ are concyclic and $\angle K'CB=\theta$. Consider another point $P$ inside $\triangle ABC$ such that $K'P\perp BC$ and $\angle PCA=\theta$. If $\sin \angle APB = \sin^2 (C-\theta)$ and the product of the lengths of the $A$- and $B$-medians of $\triangle ABC$ is $\sqrt{\sqrt{5}+1}$, then the maximum possible value of $5AB^2-CA^2-CB^2$ can be expressed in the form $m\sqrt{n}$ for positive integers $m,n$ with $n$ squarefree. Compute $100m+n$.
Let $ABC$ be a triangle with symmedian point $K$, and let $\theta = \angle AKB-90^{\circ}$. Suppose that $\theta$ is both positive and less than $\angle C$. Consider a point $K'$ inside $\triangle ABC$ such that $A,K',K,$ and $B$ are concyclic and $\angle K'CB=\theta$. Consider another point $P$ inside $\triangle ABC$ such that $K'P\perp BC$ and $\angle PCA=\theta$. If $\sin \angle APB = \sin^2 (C-\theta)$ and the product of the lengths of the $A$- and $B$-medians of $\triangle ABC$ is $\sqrt{\sqrt{5}+1}$, then the maximum possible value of $5AB^2-CA^2-CB^2$ can be expressed in the form $m\sqrt{n}$ for positive integers $m,n$ with $n$ squarefree. Compute $100m+n$.
Vincent Huang
2019 OMO Fall p2
Let $A$, $B$, $C$, and $P$ be points in the plane such that no three of them are collinear. Suppose that the areas of triangles $BPC$, $CPA$, and $APB$ are 13, 14, and 15, respectively. Compute the sum of all possible values for the area of triangle $ABC$.
Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overline{DE}$, and $\gamma_3$ is tangent to side $\overline{EF}$ and side $\overline{FA}$. Then the area of $\gamma_1$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Let $ABC$ be a triangle with incenter $I$ such that $AB=20$ and $AC=19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA=IP=IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Let $A$,$B$,$C$, and $D$ be points in the plane with $AB=AC=BC=BD=CD=36$ and such that $A \neq D$. Point $K$ lies on segment $AC$ such that $AK=2KC$. Point $M$ lies on segment $AB$, and point $N$ lies on line $AC$, such that $D$, $M$, and $N$ are collinear. Let lines $CM$ and $BN$ intersect at $P$. Then the maximum possible length of segment $KP$ can be expressed in the form $m+\sqrt{n}$ for positive integers $m$ and $n$. Compute $100m+n$.
Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that\[20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2.\]If\[\tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019,\]then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $E$ be the intersection of $BH$ and $AC$ and let $M$ and $N$ be the midpoints of $HB$ and $HO$, respectively. Let $I$ be the incenter of $AEM$ and $J$ be the intersection of $ME$ and $AI$. If $AO=20$, $AN=17$, and $\angle{ANM}=90^{\circ}$, then $\frac{AI}{AJ}=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Let $ABC$ be an acute scalene triangle with orthocenter $H$ and circumcenter $O$. Let the line through $A$ tangent to the circumcircle of triangle $AHO$ intersect the circumcircle of triangle $ABC$ at $A$ and $P \neq A$. Let the circumcircles of triangles $AOP$ and $BHP$ intersect at $P$ and $Q \neq P$. Let line $PQ$ intersect segment $BO$ at $X$. Suppose that $BX=2$, $OX=1$, and $BC=5$. Then $AB \cdot AC = \sqrt{k}+m\sqrt{n}$ for positive integers $k$, $m$, and $n$, where neither $k$ nor $n$ is divisible by the square of any integer greater than $1$. Compute $100k+10m+n$.
Let $ABC$ be a triangle. The line through $A$ tangent to the circumcircle of $ABC$ intersects line $BC$ at point $W$. Points $X,Y \neq A$ lie on lines $AC$ and $AB$, respectively, such that $WA=WX=WY$. Point $X_1$ lies on line $AB$ such that $\angle AXX_1 = 90^{\circ}$, and point $X_2$ lies on line $AC$ such that $\angle AX_1X_2 = 90^{\circ}$. Point $Y_1$ lies on line $AC$ such that $\angle AYY_1 = 90^{\circ}$, and point $Y_2$ lies on line $AB$ such that $\angle AY_1Y_2 = 90^{\circ}$. Let lines $AW$ and $XY$ intersect at point $Z$, and let point $P$ be the foot of the perpendicular from $A$ to line $X_2Y_2$. Let line $ZP$ intersect line $BC$ at $U$ and the perpendicular bisector of segment $BC$ at $V$. Suppose that $C$ lies between $B$ and $U$. Let $x$ be a positive real number. Suppose that $AB=x+1$, $AC=3$, $AV=x$, and $\frac{BC}{CU}=x$. Then $x=\frac{\sqrt{k}-m}{n}$ for positive integers $k$,$m$, and $n$ such that $k$ is not divisible by the square of any integer greater than $1$. Compute $100k+10m+n$.
2020 OMO Spring p1
Let $\ell$ be a line and let points $A$, $B$, $C$ lie on $\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $A$ perpendicular to $\ell$. Let $P$ lie on $m$. Compute the smallest possible value of $PB + PC$.
Let $ABCD$ be a square with side length $16$ and center $O$. Let $\mathcal S$ be the semicircle with diameter $AB$ that lies outside of $ABCD$, and let $P$ be a point on $\mathcal S$ so that $OP = 12$. Compute the area of triangle $CDP$.
Convex pentagon $ABCDE$ is inscribed in circle $\gamma$. Suppose that $AB=14$, $BE=10$, $BC=CD=DE$, and $[ABCDE]=3[ACD]$. Then there are two possible values for the radius of $\gamma$. The sum of these two values is $\sqrt{n}$ for some positive integer $n$. Compute $n$.
Let $ABC$ be a triangle with $AB = 20$ and $AC = 22$. Suppose its incircle touches $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at $D$, $E$, and $F$ respectively, and $P$ is the foot of the perpendicular from $D$ to $\overline{EF}$. If $\angle BPC = 90^{\circ}$, then compute $BC^2$.
Let $ABC$ be a scalene triangle. The incircle is tangent to lines $BC$, $AC$, and $AB$ at points $D$, $E$, and $F$, respectively, and the $A$-excircle is tangent to lines $BC$, $AC$, and $AB$ at points $D_1$, $E_1$, and $F_1$, respectively. Suppose that lines $AD$, $BE$, and $CF$ are concurrent at point $G$, and suppose that lines $AD_1$, $BE_1$, and $CF_1$ are concurrent at point $G_1$. Let line $GG_1$ intersect the internal bisector of angle $BAC$ at point $X$. Suppose that $AX=1$, $\cos{\angle BAC}=\sqrt{3}-1$, and $BC=8\sqrt[4]{3}$. Then $AB \cdot AC = \frac{j+k\sqrt{m}}{n}$ for positive integers $j$, $k$, $m$, and $n$ such that $\gcd(j,k,n)=1$ and $m$ is not divisible by the square of any integer greater than $1$. Compute $1000j+100k+10m+n$.
Let $ABC$ be a scalene triangle with incenter $I$ and symmedian point $K$. Furthermore, suppose that $BC = 1099$. Let $P$ be a point in the plane of triangle $ABC$, and let $D$, $E$, $F$ be the feet of the perpendiculars from $P$ to lines $BC$, $CA$, $AB$, respectively. Let $M$ and $N$ be the midpoints of segments $EF$ and $BC$, respectively. Suppose that the triples $(M,A,N)$ and $(K,I,D)$ are collinear, respectively, and that the area of triangle $DEF$ is $2020$ times the area of triangle $ABC$. Compute the largest possible value of $\lceil AB+AC\rceil$.
Let $A$, $B$ be opposite vertices of a unit square with circumcircle $\Gamma$. Let $C$ be a variable point on $\Gamma$. If $C\not\in\{A, B\}$, then let $\omega$ be the incircle of triangle $ABC$, and let $I$ be the center of $\omega$. Let $C_1$ be the point at which $\omega$ meets $\overline{AB}$, and let $D$ be the reflection of $C_1$ over line $CI$. If $C \in\{A, B\}$, let $D = C$. As $C$ varies on $\Gamma$, $D$ traces out a curve $\mathfrak C$ enclosing a region of area $\mathcal A$. Compute $\lfloor 10^4 \mathcal A\rfloor$.
Let $ABC$ be a triangle with circumcircle $\omega$ and circumcenter $O.$ Suppose that $AB = 15$, $AC = 14$, and $P$ is a point in the interior of $\triangle ABC$ such that $AP = \frac{13}{2}$, $BP^2 = \frac{409}{4}$, and $P$ is closer to $\overline{AC}$ than to $\overline{AB}$. Let $E$, $F$ be the points where $\overline{BP}$, $\overline{CP}$ intersect $\omega$ again, and let $Q$ be the intersection of $\overline{EF}$ with the tangent to $\omega$ at $A.$ Given that $AQOP$ is cyclic and that $CP^2$ is expressible in the form $\frac{a}{b} - c \sqrt{d}$ for positive integers $a$, $b$, $c$, $d$ such that $\gcd(a, b) = 1$ and $d$ is not divisible by the square of any prime, compute $1000a+100b+10c+d.$
Let $A_0BC_0D$ be a convex quadrilateral inscribed in a circle $\omega$. For all integers $i\ge0$, let $P_i$ be the intersection of lines $A_iB$ and $C_iD$, let $Q_i$ be the intersection of lines $A_iD$ and $BC_i$, let $M_i$ be the midpoint of segment $P_iQ_i$, and let lines $M_iA_i$ and $M_iC_i$ intersect $\omega$ again at $A_{i+1}$ and $C_{i+1}$, respectively. The circumcircles of $\triangle A_3M_3C_3$ and $\triangle A_4M_4C_4$ intersect at two points $U$ and $V$.
If $A_0B=3$, $BC_0=4$, $C_0D=6$, $DA_0=7$, then $UV$ can be expressed in the form $\tfrac{a\sqrt b}c$ for positive integers $a$, $b$, $c$ such that $\gcd(a,c)=1$ and $b$ is squarefree. Compute $100a+10b+c $.
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Let $A$, $B$, $C$, and $P$ be points in the plane such that no three of them are collinear. Suppose that the areas of triangles $BPC$, $CPA$, and $APB$ are 13, 14, and 15, respectively. Compute the sum of all possible values for the area of triangle $ABC$.
Ankan Bhattacharya
2019 OMO Fall p9Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overline{DE}$, and $\gamma_3$ is tangent to side $\overline{EF}$ and side $\overline{FA}$. Then the area of $\gamma_1$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Sean Li
2019 OMO Fall p11Let $ABC$ be a triangle with incenter $I$ such that $AB=20$ and $AC=19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA=IP=IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Ankit Bisain
2019 OMO Fall p15Let $A$,$B$,$C$, and $D$ be points in the plane with $AB=AC=BC=BD=CD=36$ and such that $A \neq D$. Point $K$ lies on segment $AC$ such that $AK=2KC$. Point $M$ lies on segment $AB$, and point $N$ lies on line $AC$, such that $D$, $M$, and $N$ are collinear. Let lines $CM$ and $BN$ intersect at $P$. Then the maximum possible length of segment $KP$ can be expressed in the form $m+\sqrt{n}$ for positive integers $m$ and $n$. Compute $100m+n$.
James Lin
2019 OMO Fall p16Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that\[20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2.\]If\[\tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019,\]then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Tristan Shin
2019 OMO Fall p19Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $E$ be the intersection of $BH$ and $AC$ and let $M$ and $N$ be the midpoints of $HB$ and $HO$, respectively. Let $I$ be the incenter of $AEM$ and $J$ be the intersection of $ME$ and $AI$. If $AO=20$, $AN=17$, and $\angle{ANM}=90^{\circ}$, then $\frac{AI}{AJ}=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Tristan Shin
2019 OMO Fall p24Let $ABC$ be an acute scalene triangle with orthocenter $H$ and circumcenter $O$. Let the line through $A$ tangent to the circumcircle of triangle $AHO$ intersect the circumcircle of triangle $ABC$ at $A$ and $P \neq A$. Let the circumcircles of triangles $AOP$ and $BHP$ intersect at $P$ and $Q \neq P$. Let line $PQ$ intersect segment $BO$ at $X$. Suppose that $BX=2$, $OX=1$, and $BC=5$. Then $AB \cdot AC = \sqrt{k}+m\sqrt{n}$ for positive integers $k$, $m$, and $n$, where neither $k$ nor $n$ is divisible by the square of any integer greater than $1$. Compute $100k+10m+n$.
Luke Robitaille
2019 OMO Fall p29Let $ABC$ be a triangle. The line through $A$ tangent to the circumcircle of $ABC$ intersects line $BC$ at point $W$. Points $X,Y \neq A$ lie on lines $AC$ and $AB$, respectively, such that $WA=WX=WY$. Point $X_1$ lies on line $AB$ such that $\angle AXX_1 = 90^{\circ}$, and point $X_2$ lies on line $AC$ such that $\angle AX_1X_2 = 90^{\circ}$. Point $Y_1$ lies on line $AC$ such that $\angle AYY_1 = 90^{\circ}$, and point $Y_2$ lies on line $AB$ such that $\angle AY_1Y_2 = 90^{\circ}$. Let lines $AW$ and $XY$ intersect at point $Z$, and let point $P$ be the foot of the perpendicular from $A$ to line $X_2Y_2$. Let line $ZP$ intersect line $BC$ at $U$ and the perpendicular bisector of segment $BC$ at $V$. Suppose that $C$ lies between $B$ and $U$. Let $x$ be a positive real number. Suppose that $AB=x+1$, $AC=3$, $AV=x$, and $\frac{BC}{CU}=x$. Then $x=\frac{\sqrt{k}-m}{n}$ for positive integers $k$,$m$, and $n$ such that $k$ is not divisible by the square of any integer greater than $1$. Compute $100k+10m+n$.
Ankit Bisain, Luke Robitaille, and Brandon Wang
2020 OMO Spring p1
Let $\ell$ be a line and let points $A$, $B$, $C$ lie on $\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $A$ perpendicular to $\ell$. Let $P$ lie on $m$. Compute the smallest possible value of $PB + PC$.
Ankan Bhattacharya and Brandon Wang
2020 OMO Spring p4Let $ABCD$ be a square with side length $16$ and center $O$. Let $\mathcal S$ be the semicircle with diameter $AB$ that lies outside of $ABCD$, and let $P$ be a point on $\mathcal S$ so that $OP = 12$. Compute the area of triangle $CDP$.
Brandon Wang
2020 OMO Spring p12 Convex pentagon $ABCDE$ is inscribed in circle $\gamma$. Suppose that $AB=14$, $BE=10$, $BC=CD=DE$, and $[ABCDE]=3[ACD]$. Then there are two possible values for the radius of $\gamma$. The sum of these two values is $\sqrt{n}$ for some positive integer $n$. Compute $n$.
Luke Robitaille
2020 OMO Spring p15 Let $ABC$ be a triangle with $AB = 20$ and $AC = 22$. Suppose its incircle touches $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at $D$, $E$, and $F$ respectively, and $P$ is the foot of the perpendicular from $D$ to $\overline{EF}$. If $\angle BPC = 90^{\circ}$, then compute $BC^2$.
Ankan Bhattacharya
2020 OMO Spring p19 Let $ABC$ be a scalene triangle. The incircle is tangent to lines $BC$, $AC$, and $AB$ at points $D$, $E$, and $F$, respectively, and the $A$-excircle is tangent to lines $BC$, $AC$, and $AB$ at points $D_1$, $E_1$, and $F_1$, respectively. Suppose that lines $AD$, $BE$, and $CF$ are concurrent at point $G$, and suppose that lines $AD_1$, $BE_1$, and $CF_1$ are concurrent at point $G_1$. Let line $GG_1$ intersect the internal bisector of angle $BAC$ at point $X$. Suppose that $AX=1$, $\cos{\angle BAC}=\sqrt{3}-1$, and $BC=8\sqrt[4]{3}$. Then $AB \cdot AC = \frac{j+k\sqrt{m}}{n}$ for positive integers $j$, $k$, $m$, and $n$ such that $\gcd(j,k,n)=1$ and $m$ is not divisible by the square of any integer greater than $1$. Compute $1000j+100k+10m+n$.
Luke Robitaille and Brandon Wang
2020 OMO Spring p22 Let $ABC$ be a scalene triangle with incenter $I$ and symmedian point $K$. Furthermore, suppose that $BC = 1099$. Let $P$ be a point in the plane of triangle $ABC$, and let $D$, $E$, $F$ be the feet of the perpendiculars from $P$ to lines $BC$, $CA$, $AB$, respectively. Let $M$ and $N$ be the midpoints of segments $EF$ and $BC$, respectively. Suppose that the triples $(M,A,N)$ and $(K,I,D)$ are collinear, respectively, and that the area of triangle $DEF$ is $2020$ times the area of triangle $ABC$. Compute the largest possible value of $\lceil AB+AC\rceil$.
Brandon Wang
2020 OMO Spring p24 Let $A$, $B$ be opposite vertices of a unit square with circumcircle $\Gamma$. Let $C$ be a variable point on $\Gamma$. If $C\not\in\{A, B\}$, then let $\omega$ be the incircle of triangle $ABC$, and let $I$ be the center of $\omega$. Let $C_1$ be the point at which $\omega$ meets $\overline{AB}$, and let $D$ be the reflection of $C_1$ over line $CI$. If $C \in\{A, B\}$, let $D = C$. As $C$ varies on $\Gamma$, $D$ traces out a curve $\mathfrak C$ enclosing a region of area $\mathcal A$. Compute $\lfloor 10^4 \mathcal A\rfloor$.
Brandon Wang
2020 OMO Spring p26 Let $ABC$ be a triangle with circumcircle $\omega$ and circumcenter $O.$ Suppose that $AB = 15$, $AC = 14$, and $P$ is a point in the interior of $\triangle ABC$ such that $AP = \frac{13}{2}$, $BP^2 = \frac{409}{4}$, and $P$ is closer to $\overline{AC}$ than to $\overline{AB}$. Let $E$, $F$ be the points where $\overline{BP}$, $\overline{CP}$ intersect $\omega$ again, and let $Q$ be the intersection of $\overline{EF}$ with the tangent to $\omega$ at $A.$ Given that $AQOP$ is cyclic and that $CP^2$ is expressible in the form $\frac{a}{b} - c \sqrt{d}$ for positive integers $a$, $b$, $c$, $d$ such that $\gcd(a, b) = 1$ and $d$ is not divisible by the square of any prime, compute $1000a+100b+10c+d.$
Edward Wan
2020 OMO Spring p28 Let $A_0BC_0D$ be a convex quadrilateral inscribed in a circle $\omega$. For all integers $i\ge0$, let $P_i$ be the intersection of lines $A_iB$ and $C_iD$, let $Q_i$ be the intersection of lines $A_iD$ and $BC_i$, let $M_i$ be the midpoint of segment $P_iQ_i$, and let lines $M_iA_i$ and $M_iC_i$ intersect $\omega$ again at $A_{i+1}$ and $C_{i+1}$, respectively. The circumcircles of $\triangle A_3M_3C_3$ and $\triangle A_4M_4C_4$ intersect at two points $U$ and $V$.
If $A_0B=3$, $BC_0=4$, $C_0D=6$, $DA_0=7$, then $UV$ can be expressed in the form $\tfrac{a\sqrt b}c$ for positive integers $a$, $b$, $c$ such that $\gcd(a,c)=1$ and $b$ is squarefree. Compute $100a+10b+c $.
Eric Shen
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