NIMO Monthly Contests 2012-17 65p

geometry problems from National Internet Math Olympiads (NIMO) Monthly Contests                     
with aops links in the names

NIMO Summer - Winter Constests

                                          2012 -2017 

lasted only these years

2012

2012 NIMO Mothly Contest  day 1 p4

In $\triangle ABC$, $AB = AC$. Its circumcircle, $\Gamma$, has a radius of 2. Circle $\Omega$ has a radius of 1 and is tangent to $\Gamma$, $\overline{AB}$, and $\overline{AC}$. The area of $\triangle ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$, where $b$ is squarefree and $\gcd (a, c) = 1$. Compute $a + b + c$.

Proposed by Aaron Lin 

2012 NIMO Mothly Contest  day 1 p6
A square is called proper if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by  $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. Compute $\lfloor a \rfloor$, the greatest integer less than or equal to $a$.

Proposed by Lewis Chen

2012 NIMO Mothly Contest  day 1 p7

Point $P$ lies in the interior of rectangle $ABCD$ such that $AP + CP = 27$, $BP - DP = 17$, and $\angle DAP \cong \angle DCP$. Compute the area of rectangle $ABCD$.

Proposed by Aaron Lin

2012 NIMO Mothly Contest  day 2 p3
A polygon $A_1A_2A_3\dots A_n$ is called beautiful if there exist indices $i$, $j$, and $k$ such that $\measuredangle A_iA_jA_k = 144^\circ$. Compute the number of integers $3 \le n \le 2012$ for which a regular $n$-gon is beautiful.

Proposed by Aaron Lin

2012 NIMO Mothly Contest  day 2 p5

In $\triangle ABC$, $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$, and the pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$. Compute the length of the shortest altitude of $\triangle PQR$.

Proposed by Lewis Chen 

2012 NIMO Mothly Contest  day 2 p6

In $\triangle ABC$ with circumcenter $O$, $\measuredangle A = 45^\circ$. Denote by $X$ the second intersection of $\overrightarrow{AO}$ with the circumcircle of $\triangle BOC$. Compute the area of quadrilateral $ABXC$ if $BX = 8$ and $CX = 15$.

Proposed by Aaron Lin 

2012 NIMO Mothly Contest  day 3 p1

Hexagon $ABCDEF$ is inscribed in a circle. If $\measuredangle ACE = 35^{\circ}$ and $\measuredangle CEA = 55^{\circ}$, then compute the sum of the degree measures of $\angle ABC$ and $\angle EFA$.

Proposed by Isabella Grabski 

2012 NIMO Mothly Contest  day 3 p6

In rhombus $NIMO$, $MN = 150\sqrt{3}$ and $\measuredangle MON = 60^{\circ}$. Denote by $S$ the locus of points $P$ in the interior of $NIMO$ such that $\angle MPO \cong \angle NPO$. Find the greatest integer not exceeding the perimeter of $S$.

Proposed by Evan Chen 

2012 NIMO Mothly Contest  day 3 p8

Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$.

Proposed by Lewis Chen 

2012 NIMO Mothly Contest  day 4 p7

In quadrilateral $ABCD$, $AC = BD$ and $\measuredangle B = 60^\circ$. Denote by $M$ and $N$ the midpoints of $\overline{AB}$ and $\overline{CD}$, respectively. If $MN = 12$ and the area of quadrilateral $ABCD$ is 420, then compute $AC$.

Proposed by Aaron Lin 

2012 NIMO Mothly Contest  day 4 p10

In cyclic quadrilateral $ABXC$, $\measuredangle XAB = \measuredangle XAC$. Denote by $I$ the incenter of $\triangle ABC$ and by $D$ the projection of $I$ on $\overline{BC}$. If $AI = 25$, $ID = 7$, and $BC = 14$, then $XI$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.

Proposed by Aaron Lin 

2013

2013 NIMO Mothly Contest  day 5 p3

In triangle $ABC$, $AB=13$, $BC=14$ and $CA=15$. Segment $BC$ is split into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from $A$. Find the smallest possible value of $n$.

Proposed by Evan Chen

2013 NIMO Mothly Contest  day 5 p8

Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$. Suppose that $AZ-AX=6$, $BX-BZ=9$, $AY=12$, and $BY=5$. Find the greatest integer not exceeding the perimeter of quadrilateral $OXYZ$, where $O$ is the midpoint of $AB$.

Proposed by Evan Chen

2013 NIMO Mothly Contest  day 6 p7

In $\triangle ABC$ with $AB=10$, $AC=13$, and $\measuredangle ABC = 30^\circ$, $M$ is the midpoint of $\overline{BC}$ and the circle with diameter $\overline{AM}$ meets $\overline{CB}$ and $\overline{CA}$ again at $D$ and $E$, respectively. The area of $\triangle DEM$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Compute $100m + n$.

Based on a proposal by Matthew Babbitt

2013 NIMO Mothly Contest  day 6 p9

Let $ABCD$ be a square of side length $6$. Points $E$ and $F$ are selected on rays $AB$ and $AD$ such that segments $EF$ and $BC$ intersect at a point $L$, $D$ lies between $A$ and $F$, and the area of $\triangle AEF$ is 36. Clio constructs triangle $PQR$ with $PQ=BL$, $QR=CL$ and $RP=DF$, and notices that the area of $\triangle PQR$ is $\sqrt{6}$. If the sum of all possible values of $DF$ is $\sqrt{m} + \sqrt{n}$ for positive integers $m \ge n$, compute $100m+n$.

Based on a proposal by Calvin Lee

2013 NIMO Mothly Contest  day 7 p2

In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}, \overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, $PF = 2$.

Proposed by Eugene Chen

2013 NIMO Mothly Contest  day 8 p4

On side $\overline{AB}$ of square $ABCD$, point $E$ is selected. Points $F$ and $G$ are located on sides $\overline{AB}$ and $\overline{AD}$, respectively, such that $\overline{FG} \perp \overline{CE}$. Let $P$ be the intersection point of segments $\overline{FG}$ and $\overline{CE}$. Given that $[EPF] = 1$, $[EPGA] = 8$, and $[CPFB] = 15$, compute $[PGDC]$. (Here $[\mathcal P]$ denotes the area of the polygon $\mathcal P$.)

Proposed by Aaron Lin

2013 NIMO Mothly Contest  day 8 p8

The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.

Proposed by Evan Chen

2013 NIMO Mothly Contest  day 9 p6

Let $ABC$ be a triangle with $AB = 42$, $AC = 39$, $BC = 45$. Let $E$, $F$ be on the sides $\overline{AC}$ and $\overline{AB}$ such that $AF = 21, AE = 13$. Let $\overline{CF}$ and $\overline{BE}$ intersect at $P$, and let ray $AP$ meet $\overline{BC}$ at $D$. Let $O$ denote the circumcenter of $\triangle DEF$, and $R$ its circumradius. Compute $CO^2-R^2$.

Proposed by Yang Liu

2013 NIMO Mothly Contest  day 9 p8
Let $ABCD$ be a convex quadrilateral with $\angle ABC = 120^{\circ}$ and $\angle BCD = 90^{\circ}$, and let $M$ and $N$ denote the midpoints of $\overline{BC}$ and $\overline{CD}$. Suppose there exists a point $P$ on the circumcircle of $\triangle CMN$ such that ray $MP$ bisects $\overline{AD}$ and ray $NP$ bisects $\overline{AB}$. If $AB + BC = 444$, $CD = 256$ and $BC = \frac mn$ for some relatively prime positive integers $m$ and $n$, compute $100m+n$.

Proposed by Michael Ren

2013 NIMO Mothly Contest  day 10 p7

Let $ABCD$ be a convex quadrilateral for which $DA = AB$ and $CA = CB$. Set $I_0 = C$ and $J_0 = D$, and for each nonnegative integer $n$, let $I_{n+1}$ and $J_{n+1}$ denote the incenters of $\triangle I_nAB$ and $\triangle J_nAB$, respectively.
Suppose that $$\angle DAC = 15^{\circ}, \quad \angle BAC = 65^{\circ} \quad \text{and} \quad \angle J_{2013}J_{2014}I_{2014} = \left( 90 + \frac{2k+1}{2^n} \right)^{\circ}$$ for some nonnegative integers $n$ and $k$. Compute $n+k$.

Proposed by Evan Chen

2014 

2014 NIMO Mothly Contest  day 11 p5

In triangle $ABC$, $\sin A \sin B \sin C = \frac{1}{1000}$ and $AB \cdot BC \cdot CA = 1000$. What is the area of triangle $ABC$?

Proposed by Evan Chen

2014 NIMO Mothly Contest  day 11 p8

The side lengths of $\triangle ABC$ are integers with no common factor greater than $1$. Given that $\angle B  = 2 \angle C$ and $AB < 600$, compute the sum of all possible values of $AB$.

Proposed by Eugene Chen

2014 NIMO Mothly Contest  day 12 p2

Let $ABC$ be an equilateral triangle. Denote by $D$ the midpoint of $\overline{BC}$, and denote the circle with diameter $\overline{AD}$ by $\Omega$. If the region inside $\Omega$ and outside $\triangle ABC$ has area $800\pi-600\sqrt3$, find the length of $AB$.

Proposed by Eugene Chen

2014 NIMO Mothly Contest  day 12 p5

Triangle $ABC$ has sidelengths $AB = 14, BC = 15,$ and $CA = 13$. We draw a circle with diameter $AB$ such that it passes $BC$ again at $D$ and passes $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\tfrac{m}{n}$ where $m, n$ are coprime positive integers, determine $100m+n$.

Proposed by Lewis Chen

2014 NIMO Mothly Contest  day 13 p8

Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$.

Proposed by Akshaj

2014 NIMO Mothly Contest  day 14 p1

Let $A$, $B$, $C$, $D$ be four points on a line in this order. Suppose that $AC = 25$, $BD = 40$, and $AD = 57$. Compute $AB \cdot CD + AD \cdot BC$.

Proposed by Evan Chen

2014 NIMO Mothly Contest  day 14 p3

In triangle $ABC$, we have $AB=AC=20$ and $BC=14$. Consider points $M$ on $\overline{AB}$ and $N$ on $\overline{AC}$. If the minimum value of the sum $BN + MN + MC$ is $x$, compute $100x$.

Proposed by Lewis Chen

2014 NIMO Mothly Contest  day 15 p1

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the point inside triangle $ABC$ with the property that $\overline{BD} \perp \overline{CD}$ and $\overline{AD} \perp \overline{BC}$. Then the length $AD$ can be expressed in the form $m-\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$.

Proposed by Michael Ren

2014 NIMO Mothly Contest  day 15 p4

Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$.

Proposed by Tony Kim

2014 NIMO Mothly Contest  day 16 p3

Let $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be written in the form $\tfrac pq$ where $p$ and $q$ are positive relatively prime integers. What is $100p+q$?

Proposed by David Altizio

2014 NIMO Mothly Contest  day 16 p7

Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\tfrac m{\sqrt n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$.

Proposed by David Altizio

2015 

2015 NIMO Mothly Contest  day 17 p2

Let $ABCD$ be a square with side length $100$. Denote by $M$ the midpoint of $AB$. Point $P$ is selected inside the square so that $MP = 50$ and $PC = 100$. Compute $AP^2$.

Based on a proposal by Amogh Gaitonde

2015 NIMO Mothly Contest  day 17 p6

Let $\triangle ABC$ be a triangle with $BC = 4, CA= 5, AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$.

Proposed by Sreejato Bhattacharya

2015 NIMO Mothly Contest  day 18 p3

Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \sqrt 3$. We construct four semicircles $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ whose diameters are the segments $AB$, $BC$, $CD$, $DA$. It is given that $\omega_i$ and $\omega_{i+1}$ intersect at some point $X_i$ in the interior of $ABCD$ for every $i=1,2,3,4$ (indices taken modulo $4$). Compute the square of the area of $X_1X_2X_3X_4$.

Proposed by Evan Chen

2015 NIMO Mothly Contest  day 18 p6

Let $ABC$ be a triangle with $AB=5$, $BC=7$, and $CA=8$. Let $D$ be a point on $BC$, and define points $B'$ and $C'$ on line $AD$ (or its extension) such that $BB'\perp AD$ and $CC'\perp AD$. If $B'A=B'C'$, then the ratio $BD:DC$ can be expressed in the form $m:n$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.

Proposed by Michael Ren

2015 NIMO Mothly Contest  day 18 p8

Let $ABC$ be a non-degenerate triangle with incenter $I$ and circumcircle $\Gamma$. Denote by $M_a$ the midpoint of the arc $\widehat{BC}$ of $\Gamma$ not containing $A$, and define $M_b$, $M_c$ similarly. Suppose $\triangle ABC$ has inradius $4$ and circumradius $9$. Compute the maximum possible value of $IM_a^2+IM_b^2+IM_c^2.$

Proposed by David Altizio

2015 NIMO Mothly Contest  day 19 p1
Let $\Omega_1$ and $\Omega_2$ be two circles in the plane. Suppose the common external tangent to $\Omega_1$ and $\Omega_2$ has length $2017$ while their common internal tangent has length $2009$. Find the product of the radii of $\Omega_1$ and $\Omega_2$.

Proposed by David Altizio

2015 NIMO Mothly Contest  day 19 p3

Let $O$, $A$, $B$, and $C$ be points in space such that $\angle AOB=60^{\circ}$, $\angle BOC=90^{\circ}$, and $\angle COA=120^{\circ}$. Let $\theta$ be the acute angle between planes $AOB$ and $AOC$. Given that $\cos^2\theta=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.

Proposed by Michael Ren

2015 NIMO Mothly Contest  day 19 p

Let $A_0A_1 \dots A_{11}$ be a regular $12$-gon inscribed in a circle with diameter $1$. For how many subsets $S \subseteq \{1,\dots,11\}$ is the product $$\prod_{s \in S} A_0A_s$$ equal to a rational number? (The empty product is declared to be $1$.)

Proposed by Evan Chen

2016 

2016 NIMO Mothly Contest  day 20 p6

Let $ABC$ be a triangle with $AB=20$, $AC=34$, and $BC=42$. Let $\omega_1$ and $\omega_2$ be the semicircles with diameters $\overline{AB}$ and $\overline{AC}$ erected outwards of $\triangle ABC$ and denote by $\ell$ the common external tangent to $\omega_1$ and $\omega_2$. The line through $A$ perpendicular to $\overline{BC}$ intersects $\ell$ at $X$ and $BC$ at $Y$. The length of $\overline{XY}$ can be written in the form $m+\sqrt n$ where $m$ and $n$ are positive integers. Find $100m+n$.

Proposed by David Altizio

2016 NIMO Mothly Contest  day 21 p1

In triangle $ABC,$ $AB = 13,$ $BC = 14,$ and $CA = 15.$ A circle of radius $r$ passes through point $A$ and is tangent to line $BC$ at $C.$ If $r = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $100m + n.$

Proposed by Michael Tang

2016 NIMO Mothly Contest  day 21 p6

Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$ and $BC>AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt n,$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $100m+n$.

Proposed by David Altizio and Evan Chen

2016 NIMO Mothly Contest  day 21 p8

Triangle $ABC$ has $AB=25$, $AC=29$, and $BC=36$. Additionally, $\Omega$ and $\omega$ are the circumcircle and incircle of $\triangle ABC$. Point $D$ is situated on $\Omega$ such that $AD$ is a diameter of $\Omega$, and line $AD$ intersects $\omega$ in two distinct points $X$ and $Y$. Compute $XY^2$.

Proposed by David Altizio

2016 NIMO Mothly Contest  day 22 p3

Right triangle $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $ABC$ has radius $5$. The largest possible value of $BC$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are both positive integers. Find $100m + n$.

Proposed by Jason Xia

2016 NIMO Mothly Contest  day 22 p4

In rhombus $ABCD$, let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. If $CN = 7$ and $DM = 24$, compute $AB^2$.

Proposed by Andy Liu

2016 NIMO Mothly Contest  day 23 p3

Convex pentagon $ABCDE$ satisfies $AB \parallel DE$, $BE \parallel CD$, $BC \parallel AE$, $AB = 30$, $BC = 18$, $CD = 17$, and $DE = 20$. Find its area.

Proposed by Michael Tang

2016 NIMO Mothly Contest  day 23 p8

Let $\triangle ABC$ be an equilateral triangle with side length $s$ and $P$ a point in the interior of this triangle. Suppose that $PA$, $PB$, and $PC$ are the roots of the polynomial $t^3-18t^2+91t-89$. Then $s^2$ can be written in the form $m+\sqrt n$ where $m$ and $n$ are positive integers. Find $100m+n$.

Proposed by David Altizio

2016 NIMO Mothly Contest  day 24 p5

A wall made of mirrors has the shape of $\triangle ABC$, where $AB = 13$, $BC = 16$, and $CA = 9$. A laser positioned at point $A$ is fired at the midpoint $M$ of $BC$. The shot reflects about $BC$ and then strikes point $P$ on $AB$. If $\tfrac{AM}{MP} = \tfrac{m}{n}$ for relatively prime positive integers $m, n$, compute $100m+n$.

Proposed by Michael Tang

2016 NIMO Mothly Contest  day 24 p7

Let $A$ and $B$ be points with $AB=12$. A point $P$ in the plane of $A$ and $B$ is $\textit{special}$ if there exist points $X, Y$ such that
$P$ lies on segment $XY$,
$PX : PY = 4 : 7$, and
the circumcircles of $AXY$ and $BXY$ are both tangent to line $AB$.
A point $P$ that is not special is called $\textit{boring}$.
Compute the smallest integer $n$ such that any two boring points have distance less than $\sqrt{n/10}$ from each other.

Proposed by Michael Ren

2016 NIMO Mothly Contest  day 25 p1

In quadrilateral $ABCD$, $AB \parallel CD$ and $BC \perp AB$. Lines $AC$ and $BD$ intersect at $E$. If $AB = 20$, $BC = 2016$, and $CD = 16$, find the area of $\triangle BCE$.

Proposed by Harrison Wang

2016 NIMO Mothly Contest  day 25 p8

Rectangle $EFGH$ with side lengths $8$, $9$ lies inside rectangle $ABCD$ with side lengths $13$, $14$, with their corresponding sides parallel. Let $\ell_A, \ell_B, \ell_C, \ell_D$ be the lines connecting $A,B,C,D$, respectively, with the vertex of $EFGH$ closest to them. Let $P = \ell_A \cap \ell_B$, $Q = \ell_B \cap \ell_C$, $R = \ell_C \cap \ell_D$, and $S = \ell_D \cap \ell_A$. Suppose that the greatest possible area of quadrilateral $PQRS$ is $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$.

Proposed by Yannick Yao

2016 NIMO Mothly Contest  day 26 p1

Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs. If $K = \sqrt{a} - b$ for positive integers $a, b$, find $100a+b$.

Proposed by Michael Tang

2016 NIMO Mothly Contest  day 26 p4

Triangle $ABC$ has $AB=13$, $BC=14$, and $CA=15$. Let $\omega_A$, $\omega_B$ and $\omega_C$ be circles such that $\omega_B$ and $\omega_C$ are tangent at $A$, $\omega_C$ and $\omega_A$ are tangent at $B$, and $\omega_A$ and $\omega_B$ are tangent at $C$. Suppose that line $AB$ intersects $\omega_B$ at a point $X \neq A$ and line $AC$ intersects $\omega_C$ at a point $Y \neq A$. If lines $XY$ and $BC$ intersect at $P$, then $\tfrac{BC}{BP} = \tfrac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m+n$.

Proposed by Michael Ren

2017 

2017 NIMO Mothly Contest  day 27 p2

An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$, $Q\in\overline{AC}$, and $N,P\in\overline{BC}$.

Suppose that $ABC$ is an equilateral triangle of side length $2$, and that $AMNPQ$ has a line of symmetry perpendicular to $BC$. Then the area of $AMNPQ$ is $n-p\sqrt{q}$, where $n, p, q$ are positive integers and $q$ is not divisible by the square of a prime. Compute $100n+10p+q$.

Proposed by Michael Ren

2017 NIMO Mothly Contest  day 27 p6

In $\triangle ABC$, $AB = 4$, $BC = 5$, and $CA = 6$. Circular arcs $p$, $q$, $r$ of measure $60^\circ$ are drawn from $A$ to $B$, from $A$ to $C$, and from $B$ to $C$, respectively, so that $p$, $q$ lie completely outside $\triangle ABC$ but $r$ does not. Let $X$, $Y$, $Z$ be the midpoints of $p$, $q$, $r$, respectively. If $\sin \angle XZY = \dfrac{a\sqrt{b}+c}{d}$, where $a, b, c, d$ are positive integers, $\gcd(a,c,d)=1$, and $b$ is not divisible by the square of a prime, compute $a+b+c+d$.

Proposed by Michael Tang

2017 NIMO Mothly Contest  day 28 p2

Trapezoid $ABCD$ is an isosceles trapezoid with $AD=BC$. Point $P$ is the intersection of the diagonals $AC$ and $BD$. If the area of $\triangle ABP$ is $50$ and the area of $\triangle CDP$ is $72$, what is the area of the entire trapezoid?

Proposed by David Altizio

2017 NIMO Mothly Contest  day 28 p8

Let $ABC$ be a triangle with $BC=49$ and circumradius $25$. Suppose that the circle centered on $BC$ that is tangent to $AB$ and $AC$ is also tangent to the circumcircle of $ABC$. Then $$\dfrac{AB \cdot AC}{-BC+AB+AC} = \frac{m}{n}$$ where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.

Proposed by Michael Ren

2017 NIMO Mothly Contest  day 29 p3

A circle $C_0$ is inscribed in an equilateral triangle $XYZ$ of side length 112. Then, for each positive integer $n$, circle $C_n$ is inscribed in the region bounded by $XY$, $XZ$, and an arc of circle $C_{n-1}$, forming an infinite sequence of circles tangent to sides $XY$ and $XZ$ and approaching vertex $X$. If these circles collectively have area $m\pi$, find $m$.

Proposed by Michael Tang

2017 NIMO Mothly Contest  day 29 p6

Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC =60^\circ$ and $O$ lies on $\gamma$. If $$\tan B \tan C = a + \sqrt{b}$$ for positive integers $a$ and $b$, compute $100a+b$.

Proposed by Kaan Dokmeci

2017 NIMO Mothly Contest  day 30 p5

In triangle $ABC$, $AB=12$, $BC=17$, and $AC=25$. Distinct points $M$ and $N$ lie on the circumcircle of $ABC$ such that $BM=CM$ and $BN=CN$. If $AM + AN = \tfrac{a\sqrt{b}}{c}$, where $a, b, c$ are positive integers such that $\gcd(a, c) = 1$ and $b$ is not divisible by the square of a prime, compute $100a+10b+c$.

Proposed by Michael Tang

2017 NIMO Mothly Contest  day 30 p7

Let $ABC$ be a triangle with $AB=4$, $AC=5$, $BC=6$, and circumcircle $\Omega$. Points $E$ and $F$ lie on $AC$ and $AB$ respectively such that $\angle ABE=\angle CBE$ and $\angle ACF=\angle BCF$. The second intersection point of the circumcircle of $\triangle AEF$ with $\Omega$ (other than $A$) is $P$. Suppose $AP^2=\frac mn$ where $m$ and $n$ are positive relatively prime integers. Find $100m+n$.

Proposed by David Altizio

2017 NIMO Mothly Contest  day 31 p3

Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$.

Proposed by Anand Iyer

2017 NIMO Mothly Contest  day 31 p5

Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Points $D$ and $E$ are chosen on $AC$ and $AB$, respectively, such that quadrilateral $BCDE$ is cyclic and when the triangle is folded along segment $DE$, point $A$ lies on side $BC$. If the length of $DE$ can be expressed as $\tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $100m+n$.

Proposd by Joseph Heerens

2017 NIMO Mothly Contest  day 32 p3

In rectangle $ABCD$ with center $O$, $AB=10$ and $BC=8$. Circle $\gamma$ has center $O$ and lies tangent to $\overline{AB}$ and $\overline{CD}$. Points $M$ and $N$ are chosen on $\overline{AD}$ and  $\overline{BC}$, respectively; segment $MN$ intersects $\gamma$ at two distinct points $P$ and $Q$, with $P$ between $M$ and $Q$. If $MP : PQ : QN = 3 : 5 : 2$, then the length $MN$ can be expressed in the form $\sqrt{a} - \sqrt{b}$, where $a$, $b$ are positive integers. Find $100a + b$.

Proposed by Michael Tang

source:

http://internetolympiad.org