geometry problems from Carnegie Mellon Informatics and Mathematics Competition (CMIMC)
with aops links in the names
2016 - 2021
2016 CMIMC Geometry 1
Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.
2016 CMIMC Geometry 2
Let $ABCD$ be an isosceles trapezoid with $AD=BC=15$ such that the distance between its bases $AB$ and $CD$ is $7$. Suppose further that the circles with diameters $\overline{AD}$ and $\overline{BC}$ are tangent to each other. What is the area of the trapezoid?
2016 CMIMC Geometry Tiebraker 1
Point $A$ lies on the circumference of a circle $\Omega$ with radius $78$. Point $B$ is placed such that $AB$ is tangent to the circle and $AB=65$, while point $C$ is located on $\Omega$ such that $BC=25$. Compute the length of $\overline{AC}$.
2016 CMIMC Geometry Tiebraker 2
Identical spherical tennis balls of radius 1 are placed inside a cylindrical container of radius 2 and height 19. Compute the maximum number of tennis balls that can fit entirely inside this container.
2016 CMIMC Team 2
Right isosceles triangle $T$ is placed in the first quadrant of the coordinate plane. Suppose that the projection of $T$ onto the $x$-axis has length $6$, while the projection of $T$ onto the $y$-axis has length $8$. What is the sum of all possible areas of the triangle $T$?
collected inside aops here
2016 - 2021
2016 CMIMC Geometry 1
Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.
2016 CMIMC Geometry 2
Let $ABCD$ be an isosceles trapezoid with $AD=BC=15$ such that the distance between its bases $AB$ and $CD$ is $7$. Suppose further that the circles with diameters $\overline{AD}$ and $\overline{BC}$ are tangent to each other. What is the area of the trapezoid?
2016 CMIMC Geometry 3
Let $ABC$ be a triangle. The angle bisector of $\angle B$ intersects $AC$ at point $P$, while the angle bisector of $\angle C$ intersects $AB$ at a point $Q$. Suppose the area of $\triangle ABP$ is 27, the area of $\triangle ACQ$ is 32, and the area of $\triangle ABC$ is $72$. The length of $\overline{BC}$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers with $n$ as small as possible. What is $m+n$?
2016 CMIMC Geometry 4
Andrew the Antelope canters along the surface of a regular icosahedron, which has twenty equilateral triangle faces and edge length 4. If he wants to move from one vertex to the opposite vertex, the minimum distance he must travel can be expressed as $\sqrt{n}$ for some integer $n$. Compute $n$.
Let $ABC$ be a triangle. The angle bisector of $\angle B$ intersects $AC$ at point $P$, while the angle bisector of $\angle C$ intersects $AB$ at a point $Q$. Suppose the area of $\triangle ABP$ is 27, the area of $\triangle ACQ$ is 32, and the area of $\triangle ABC$ is $72$. The length of $\overline{BC}$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers with $n$ as small as possible. What is $m+n$?
2016 CMIMC Geometry 4
Andrew the Antelope canters along the surface of a regular icosahedron, which has twenty equilateral triangle faces and edge length 4. If he wants to move from one vertex to the opposite vertex, the minimum distance he must travel can be expressed as $\sqrt{n}$ for some integer $n$. Compute $n$.
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2+y^2+z^2$.
2016 CMIMC Geometry 6
In parallelogram $ABCD$, angles $B$ and $D$ are acute while angles $A$ and $C$ are obtuse. The perpendicular from $C$ to $AB$ and the perpendicular from $A$ to $BC$ intersect at a point $P$ inside the parallelogram. If $PB=700$ and $PD=821$, what is $AC$?
In parallelogram $ABCD$, angles $B$ and $D$ are acute while angles $A$ and $C$ are obtuse. The perpendicular from $C$ to $AB$ and the perpendicular from $A$ to $BC$ intersect at a point $P$ inside the parallelogram. If $PB=700$ and $PD=821$, what is $AC$?
2016 CMIMC Geometry 7
Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$.
Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$.
2016 CMIMC Geometry 8
Suppose $ABCD$ is a convex quadrilateral satisfying $AB=BC$, $AC=BD$, $\angle ABD = 80^\circ$, and $\angle CBD = 20^\circ$. What is $\angle BCD$ in degrees?
Suppose $ABCD$ is a convex quadrilateral satisfying $AB=BC$, $AC=BD$, $\angle ABD = 80^\circ$, and $\angle CBD = 20^\circ$. What is $\angle BCD$ in degrees?
2016 CMIMC Geometry 9
Let $\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\Gamma$ with diameter $\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\Gamma$ at a point $X$. The length $AX$ can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $m+n$.
Let $\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\Gamma$ with diameter $\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\Gamma$ at a point $X$. The length $AX$ can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $m+n$.
Let $\triangle ABC$ be a triangle with circumcircle $\Omega$ and let $N$ be the midpoint of the major arc $\widehat{BC}$. The incircle $\omega$ of $\triangle ABC$ is tangent to $AC$ and $AB$ at points $E$ and $F$ respectively. Suppose point $X$ is placed on the same side of $EF$ as $A$ such that $\triangle XEF\sim\triangle ABC$. Let $NX$ intersect $BC$ at a point $P$. Given that $AB=15$, $BC=16$, and $CA=17$, compute $\tfrac{PX}{XN}$.
Point $A$ lies on the circumference of a circle $\Omega$ with radius $78$. Point $B$ is placed such that $AB$ is tangent to the circle and $AB=65$, while point $C$ is located on $\Omega$ such that $BC=25$. Compute the length of $\overline{AC}$.
2016 CMIMC Geometry Tiebraker 2
Identical spherical tennis balls of radius 1 are placed inside a cylindrical container of radius 2 and height 19. Compute the maximum number of tennis balls that can fit entirely inside this container.
Triangle $ABC$ satisfies $AB=28$, $BC=32$, and $CA=36$, and $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. Let point $P$ be the unique point in the plane $ABC$ such that $\triangle PBM\sim\triangle PNC$. What is $AP$?
Right isosceles triangle $T$ is placed in the first quadrant of the coordinate plane. Suppose that the projection of $T$ onto the $x$-axis has length $6$, while the projection of $T$ onto the $y$-axis has length $8$. What is the sum of all possible areas of the triangle $T$?
2016 CMIMC Team 7
In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.
In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.
2016 CMIMC Team 10
Let $\mathcal{P}$ be the unique parabola in the $xy$-plane which is tangent to the $x$-axis at $(5,0)$ and to the $y$-axis at $(0,12)$. We say a line $\ell$ is $\mathcal{P}$-friendly if the $x$-axis, $y$-axis, and $\mathcal{P}$ divide $\ell$ into three segments, each of which has equal length. If the sum of the slopes of all $\mathcal{P}$-friendly lines can be written in the form $-\tfrac mn$ for $m$ and $n$ positive relatively prime integers, find $m+n$.
2017 CMIMC Geometry 1
Let $ABC$ be a triangle with $\angle BAC=117^\circ$. The angle bisector of $\angle ABC$ intersects side $AC$ at $D$. Suppose $\triangle ABD\sim\triangle ACB$. Compute the measure of $\angle ABC$, in degrees.
2017 CMIMC Geometry Tiebraker 1
Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP\perp CD$ and $BQ\perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX=3$ and $XQ=1$. Compute the largest possible area of $ABCD$.
2017 CMIMC Geometry Tiebraker 2
2017 CMIMC Geometry Tiebraker 3
Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.
2017 CMIMC Team 9
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other. Circle $\Omega$ is placed such that $\omega_1$ is internally tangent to $\Omega$ at $X$ while $\omega_2$ is internally tangent to $\Omega$ at $Y$. Line $\ell$ is tangent to $\omega_1$ at $P$ and $\omega_2$ at $Q$ and furthermore intersects $\Omega$ at points $A$ and $B$ with $AP<AQ$. Suppose that $AP=2$, $PQ=4$, and $QB=3$. Compute the length of line segment $\overline{XY}$.
2018 CMIMC Geometry 1
Let $ABC$ be a triangle. Point $P$ lies in the interior of $\triangle ABC$ such that $\angle ABP = 20^\circ$ and $\angle ACP = 15^\circ$. Compute $\angle BPC - \angle BAC$.
2018 CMIMC Geometry 2
Let $ABCD$ be a square of side length $1$, and let $P$ be a variable point on $\overline{CD}$. Denote by $Q$ the intersection point of the angle bisector of $\angle APB$ with $\overline{AB}$. The set of possible locations for $Q$ as $P$ varies along $\overline{CD}$ is a line segment; what is the length of this segment?
2018 CMIMC Geometry 3
Let $ABC$ be a triangle with side lengths $5$, $4\sqrt 2$, and $7$. What is the area of the triangle with side lengths $\sin A$, $\sin B$, and $\sin C$?
2018 CMIMC Geometry 4
Suppose $\overline{AB}$ is a segment of unit length in the plane. Let $f(X)$ and $g(X)$ be functions of the plane such that $f$ corresponds to rotation about $A$ $60^\circ$ counterclockwise and $g$ corresponds to rotation about $B$ $90^\circ$ clockwise. Let $P$ be a point with $g(f(P))=P$; what is the sum of all possible distances from $P$ to line $AB$?
2018 CMIMC Geometry 5
Select points $T_1,T_2$ and $T_3$ in $\mathbb{R}^3$ such that $T_1=(0,1,0)$, $T_2$ is at the origin, and $T_3=(1,0,0)$. Let $T_0$ be a point on the line $x=y=0$ with $T_0\neq T_2$. Suppose there exists a point $X$ in the plane of $\triangle T_1T_2T_3$ such that the quantity $(XT_i)[T_{i+1}T_{i+2}T_{i+3}]$ is constant for all $i=0$ to $i=3$, where $[\mathcal{P}]$ denotes area of the polygon $\mathcal{P}$ and indices are taken modulo 4. What is the magnitude of the $z$-coordinate of $T_0$?
2018 CMIMC Geometry 6
Let $\omega_1$ and $\omega_2$ be intersecting circles in the plane with radii $12$ and $15$, respectively. Suppose $\Gamma$ is a circle such that $\omega_1$ and $\omega_2$ are internally tangent to $\Gamma$ at $X_1$ and $X_2$, respectively. Similarly, $\ell$ is a line that is tangent to $\omega_1$ and $\omega_2$ at $Y_1$ and $Y_2$, respectively. If $X_1X_2=18$ and $Y_1Y_2=9$, what is the radius of $\Gamma$?
2018 CMIMC Geometry 7
Let $ABC$ be a triangle with $AB=10$, $AC=11$, and circumradius $6$. Points $D$ and $E$ are located on the circumcircle of $\triangle ABC$ such that $\triangle ADE$ is equilateral. Line segments $\overline{DE}$ and $\overline{BC}$ intersect at $X$. Find $\tfrac{BX}{XC}$.
2018 CMIMC Geometry 8
In quadrilateral $ABCD$, $AB=2$, $AD=3$, $BC=CD=\sqrt7$, and $\angle DAB=60^\circ$. Semicircles $\gamma_1$ and $\gamma_2$ are erected on the exterior of the quadrilateral with diameters $\overline{AB}$ and $\overline{AD}$; points $E\neq B$ and $F\neq D$ are selected on $\gamma_1$ and $\gamma_2$ respectively such that $\triangle CEF$ is equilateral. What is the area of $\triangle CEF$?
2018 CMIMC Geometry 9
Suppose $\mathcal{E}_1 \neq \mathcal{E}_2$ are two intersecting ellipses with a common focus $X$; let the common external tangents of $\mathcal{E}_1$ and $\mathcal{E}_2$ intersect at a point $Y$. Further suppose that $X_1$ and $X_2$ are the other foci of $\mathcal{E}_1$ and $\mathcal{E}_2$, respectively, such that $X_1\in \mathcal{E}_2$ and $X_2\in \mathcal{E}_1$. If $X_1X_2=8, XX_2=7$, and $XX_1=9$, what is $XY^2$?
2018 CMIMC Geometry 10
Let $ABC$ be a triangle with circumradius $17$, inradius $4$, circumcircle $\Gamma$ and $A$-excircle $\Omega$. Suppose the reflection of $\Omega$ over line $BC$ is internally tangent to $\Gamma$. Compute the area of $\triangle ABC$.
2018 CMIMC Geometry Tiebraker 1
Let $ABC$ be a triangle with $AB=9$, $BC=10$, $CA=11$, and orthocenter $H$. Suppose point $D$ is placed on $\overline{BC}$ such that $AH=HD$. Compute $AD$.
2018 CMIMC Team 1.1
Let $ABC$ be a triangle with $BC=30$, $AC=50$, and $AB=60$. Circle $\omega_B$ is the circle passing through $A$ and $B$ tangent to $BC$ at $B$; $\omega_C$ is defined similarly. Suppose the tangent to $\odot(ABC)$ at $A$ intersects $\omega_B$ and $\omega_C$ for the second time at $X$ and $Y$ respectively. Compute $XY$.
Let $T = TNYWR$. Let $CMU$ be a triangle with $CM=13$, $MU=14$, and $UC=15$. Rectangle $WEAN$ is inscribed in $\triangle CMU$ with points $W$ and $E$ on $\overline{MU}$, point $A$ on $\overline{CU}$, and point $N$ on $\overline{CM}$. If the area of $WEAN$ is $T$, what is its perimeter?
Let $\mathcal{P}$ be the unique parabola in the $xy$-plane which is tangent to the $x$-axis at $(5,0)$ and to the $y$-axis at $(0,12)$. We say a line $\ell$ is $\mathcal{P}$-friendly if the $x$-axis, $y$-axis, and $\mathcal{P}$ divide $\ell$ into three segments, each of which has equal length. If the sum of the slopes of all $\mathcal{P}$-friendly lines can be written in the form $-\tfrac mn$ for $m$ and $n$ positive relatively prime integers, find $m+n$.
2017 CMIMC Geometry 1
Let $ABC$ be a triangle with $\angle BAC=117^\circ$. The angle bisector of $\angle ABC$ intersects side $AC$ at $D$. Suppose $\triangle ABD\sim\triangle ACB$. Compute the measure of $\angle ABC$, in degrees.
2017 CMIMC Geometry 2
Triangle $ABC$ has an obtuse angle at $\angle A$. Points $D$ and $E$ are placed on $\overline{BC}$ in the order $B$, $D$, $E$, $C$ such that $\angle BAD=\angle BCA$ and $\angle CAE=\angle CBA$. If $AB=10$, $AC=11$, and $DE=4$, determine $BC$.
Triangle $ABC$ has an obtuse angle at $\angle A$. Points $D$ and $E$ are placed on $\overline{BC}$ in the order $B$, $D$, $E$, $C$ such that $\angle BAD=\angle BCA$ and $\angle CAE=\angle CBA$. If $AB=10$, $AC=11$, and $DE=4$, determine $BC$.
2017 CMIMC Geometry 3
In acute triangle $ABC$, points $D$ and $E$ are the feet of the angle bisector and altitude from $A$ respectively. Suppose that $AC - AB = 36$ and $DC - DB = 24$. Compute $EC - EB$.
In acute triangle $ABC$, points $D$ and $E$ are the feet of the angle bisector and altitude from $A$ respectively. Suppose that $AC - AB = 36$ and $DC - DB = 24$. Compute $EC - EB$.
2017 CMIMC Geometry 4
Let $\mathcal S$ be the sphere with center $(0,0,1)$ and radius $1$ in $\mathbb R^3$. A plane $\mathcal P$ is tangent to $\mathcal S$ at the point $(x_0,y_0,z_0)$, where $x_0$, $y_0$, and $z_0$ are all positive. Suppose the intersection of plane $\mathcal P$ with the $xy$-plane is the line with equation $2x+y=10$ in $xy$-space. What is $z_0$?
Let $\mathcal S$ be the sphere with center $(0,0,1)$ and radius $1$ in $\mathbb R^3$. A plane $\mathcal P$ is tangent to $\mathcal S$ at the point $(x_0,y_0,z_0)$, where $x_0$, $y_0$, and $z_0$ are all positive. Suppose the intersection of plane $\mathcal P$ with the $xy$-plane is the line with equation $2x+y=10$ in $xy$-space. What is $z_0$?
2017 CMIMC Geometry 5
Two circles $\omega_1$ and $\omega_2$ are said to be $\textit{orthogonal}$ if they intersect each other at right angles. In other words, for any point $P$ lying on both $\omega_1$ and $\omega_2$, if $\ell_1$ is the line tangent to $\omega_1$ at $P$ and $\ell_2$ is the line tangent to $\omega_2$ at $P$, then $\ell_1\perp \ell_2$. (Two circles which do not intersect are not orthogonal.)
Two circles $\omega_1$ and $\omega_2$ are said to be $\textit{orthogonal}$ if they intersect each other at right angles. In other words, for any point $P$ lying on both $\omega_1$ and $\omega_2$, if $\ell_1$ is the line tangent to $\omega_1$ at $P$ and $\ell_2$ is the line tangent to $\omega_2$ at $P$, then $\ell_1\perp \ell_2$. (Two circles which do not intersect are not orthogonal.)
Let $\triangle ABC$ be a triangle with area $20$. Orthogonal circles $\omega_B$ and $\omega_C$ are drawn with $\omega_B$ centered at $B$ and $\omega_C$ centered at $C$. Points $T_B$ and $T_C$ are placed on $\omega_B$ and $\omega_C$ respectively such that $AT_B$ is tangent to $\omega_B$ and $AT_C$ is tangent to $\omega_C$. If $AT_B = 7$ and $AT_C = 11$, what is $\tan\angle BAC$?
2017 CMIMC Geometry 6
Cyclic quadrilateral $ABCD$ satisfies $\angle ABD = 70^\circ$, $\angle ADB=50^\circ$, and $BC=CD$. Suppose $AB$ intersects $CD$ at point $P$, while $AD$ intersects $BC$ at point $Q$. Compute $\angle APQ-\angle AQP$.
Cyclic quadrilateral $ABCD$ satisfies $\angle ABD = 70^\circ$, $\angle ADB=50^\circ$, and $BC=CD$. Suppose $AB$ intersects $CD$ at point $P$, while $AD$ intersects $BC$ at point $Q$. Compute $\angle APQ-\angle AQP$.
2017 CMIMC Geometry 7
Two non-intersecting circles, $\omega$ and $\Omega$, have centers $C_\omega$ and $C_\Omega$ respectively. It is given that the radius of $\Omega$ is strictly larger than the radius of $\omega$. The two common external tangents of $\Omega$ and $\omega$ intersect at a point $P$, and an internal tangent of the two circles intersects the common external tangents at $X$ and $Y$. Suppose that the radius of $\omega$ is $4$, the circumradius of $\triangle PXY$ is $9$, and $XY$ bisects $\overline{PC_\Omega}$. Compute $XY$.
Two non-intersecting circles, $\omega$ and $\Omega$, have centers $C_\omega$ and $C_\Omega$ respectively. It is given that the radius of $\Omega$ is strictly larger than the radius of $\omega$. The two common external tangents of $\Omega$ and $\omega$ intersect at a point $P$, and an internal tangent of the two circles intersects the common external tangents at $X$ and $Y$. Suppose that the radius of $\omega$ is $4$, the circumradius of $\triangle PXY$ is $9$, and $XY$ bisects $\overline{PC_\Omega}$. Compute $XY$.
2017 CMIMC Geometry 8
In triangle $ABC$ with $AB=23$, $AC=27$, and $BC=20$, let $D$ be the foot of the $A$ altitude. Let $\mathcal{P}$ be the parabola with focus $A$ passing through $B$ and $C$, and denote by $T$ the intersection point of $AD$ with the directrix of $\mathcal P$. Determine the value of $DT^2-DA^2$. (Recall that a parabola $\mathcal P$ is the set of points which are equidistant from a point, called the $\textit{focus}$ of $\mathcal P$, and a line, called the $\textit{directrix}$ of $\mathcal P$.)
In triangle $ABC$ with $AB=23$, $AC=27$, and $BC=20$, let $D$ be the foot of the $A$ altitude. Let $\mathcal{P}$ be the parabola with focus $A$ passing through $B$ and $C$, and denote by $T$ the intersection point of $AD$ with the directrix of $\mathcal P$. Determine the value of $DT^2-DA^2$. (Recall that a parabola $\mathcal P$ is the set of points which are equidistant from a point, called the $\textit{focus}$ of $\mathcal P$, and a line, called the $\textit{directrix}$ of $\mathcal P$.)
2017 CMIMC Geometry 9
Let $\triangle ABC$ be an acute triangle with circumcenter $O$, and let $Q\neq A$ denote the point on $\odot (ABC)$ for which $AQ\perp BC$. The circumcircle of $\triangle BOC$ intersects lines $AC$ and $AB$ for the second time at $D$ and $E$ respectively. Suppose that $AQ$, $BC$, and $DE$ are concurrent. If $OD=3$ and $OE=7$, compute $AQ$.
Let $\triangle ABC$ be an acute triangle with circumcenter $O$, and let $Q\neq A$ denote the point on $\odot (ABC)$ for which $AQ\perp BC$. The circumcircle of $\triangle BOC$ intersects lines $AC$ and $AB$ for the second time at $D$ and $E$ respectively. Suppose that $AQ$, $BC$, and $DE$ are concurrent. If $OD=3$ and $OE=7$, compute $AQ$.
2017 CMIMC Geometry 10
Suppose $\triangle ABC$ is such that $AB=13$, $AC=15$, and $BC=14$. It is given that there exists a unique point $D$ on side $\overline{BC}$ such that the Euler lines of $\triangle ABD$ and $\triangle ACD$ are parallel. Determine the value of $\tfrac{BD}{CD}$. (The $\textit{Euler}$ line of a triangle $ABC$ is the line connecting the centroid, circumcenter, and orthocenter of $ABC$.)
Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP\perp CD$ and $BQ\perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX=3$ and $XQ=1$. Compute the largest possible area of $ABCD$.
2017 CMIMC Geometry Tiebraker 2
Points $A$, $B$, and $C$ lie on a circle $\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\ell$ tangent to the incircle of $\triangle ABC$ at $T$ intersects $\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC=40$, and $XY=48$. Compute $TX\cdot TY$.
Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.
2017 CMIMC Team 9
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other. Circle $\Omega$ is placed such that $\omega_1$ is internally tangent to $\Omega$ at $X$ while $\omega_2$ is internally tangent to $\Omega$ at $Y$. Line $\ell$ is tangent to $\omega_1$ at $P$ and $\omega_2$ at $Q$ and furthermore intersects $\Omega$ at points $A$ and $B$ with $AP<AQ$. Suppose that $AP=2$, $PQ=4$, and $QB=3$. Compute the length of line segment $\overline{XY}$.
2018 CMIMC Geometry 1
Let $ABC$ be a triangle. Point $P$ lies in the interior of $\triangle ABC$ such that $\angle ABP = 20^\circ$ and $\angle ACP = 15^\circ$. Compute $\angle BPC - \angle BAC$.
Let $ABCD$ be a square of side length $1$, and let $P$ be a variable point on $\overline{CD}$. Denote by $Q$ the intersection point of the angle bisector of $\angle APB$ with $\overline{AB}$. The set of possible locations for $Q$ as $P$ varies along $\overline{CD}$ is a line segment; what is the length of this segment?
Let $ABC$ be a triangle with side lengths $5$, $4\sqrt 2$, and $7$. What is the area of the triangle with side lengths $\sin A$, $\sin B$, and $\sin C$?
Suppose $\overline{AB}$ is a segment of unit length in the plane. Let $f(X)$ and $g(X)$ be functions of the plane such that $f$ corresponds to rotation about $A$ $60^\circ$ counterclockwise and $g$ corresponds to rotation about $B$ $90^\circ$ clockwise. Let $P$ be a point with $g(f(P))=P$; what is the sum of all possible distances from $P$ to line $AB$?
Select points $T_1,T_2$ and $T_3$ in $\mathbb{R}^3$ such that $T_1=(0,1,0)$, $T_2$ is at the origin, and $T_3=(1,0,0)$. Let $T_0$ be a point on the line $x=y=0$ with $T_0\neq T_2$. Suppose there exists a point $X$ in the plane of $\triangle T_1T_2T_3$ such that the quantity $(XT_i)[T_{i+1}T_{i+2}T_{i+3}]$ is constant for all $i=0$ to $i=3$, where $[\mathcal{P}]$ denotes area of the polygon $\mathcal{P}$ and indices are taken modulo 4. What is the magnitude of the $z$-coordinate of $T_0$?
2018 CMIMC Geometry 6
Let $\omega_1$ and $\omega_2$ be intersecting circles in the plane with radii $12$ and $15$, respectively. Suppose $\Gamma$ is a circle such that $\omega_1$ and $\omega_2$ are internally tangent to $\Gamma$ at $X_1$ and $X_2$, respectively. Similarly, $\ell$ is a line that is tangent to $\omega_1$ and $\omega_2$ at $Y_1$ and $Y_2$, respectively. If $X_1X_2=18$ and $Y_1Y_2=9$, what is the radius of $\Gamma$?
Let $ABC$ be a triangle with $AB=10$, $AC=11$, and circumradius $6$. Points $D$ and $E$ are located on the circumcircle of $\triangle ABC$ such that $\triangle ADE$ is equilateral. Line segments $\overline{DE}$ and $\overline{BC}$ intersect at $X$. Find $\tfrac{BX}{XC}$.
2018 CMIMC Geometry 8
In quadrilateral $ABCD$, $AB=2$, $AD=3$, $BC=CD=\sqrt7$, and $\angle DAB=60^\circ$. Semicircles $\gamma_1$ and $\gamma_2$ are erected on the exterior of the quadrilateral with diameters $\overline{AB}$ and $\overline{AD}$; points $E\neq B$ and $F\neq D$ are selected on $\gamma_1$ and $\gamma_2$ respectively such that $\triangle CEF$ is equilateral. What is the area of $\triangle CEF$?
Suppose $\mathcal{E}_1 \neq \mathcal{E}_2$ are two intersecting ellipses with a common focus $X$; let the common external tangents of $\mathcal{E}_1$ and $\mathcal{E}_2$ intersect at a point $Y$. Further suppose that $X_1$ and $X_2$ are the other foci of $\mathcal{E}_1$ and $\mathcal{E}_2$, respectively, such that $X_1\in \mathcal{E}_2$ and $X_2\in \mathcal{E}_1$. If $X_1X_2=8, XX_2=7$, and $XX_1=9$, what is $XY^2$?
Let $ABC$ be a triangle with circumradius $17$, inradius $4$, circumcircle $\Gamma$ and $A$-excircle $\Omega$. Suppose the reflection of $\Omega$ over line $BC$ is internally tangent to $\Gamma$. Compute the area of $\triangle ABC$.
Let $ABC$ be a triangle with $AB=9$, $BC=10$, $CA=11$, and orthocenter $H$. Suppose point $D$ is placed on $\overline{BC}$ such that $AH=HD$. Compute $AD$.
2018 CMIMC Geometry Tiebraker 2
Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB=10$ and $CD=15$, compute the maximum possible value of $XY$.
Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB=10$ and $CD=15$, compute the maximum possible value of $XY$.
Let $ABC$ be a triangle with incircle $\omega$ and incenter $I$. The circle $\omega$ is tangent to $BC$, $CA$, and $AB$ at $D$, $E$, and $F$ respectively. Point $P$ is the foot of the angle bisector from $A$ to $BC$, and point $Q$ is the foot of the altitude from $D$ to $EF$. Suppose $AI=7$, $IP=5$, and $DQ=4$. Compute the radius of $\omega$.
Let $ABC$ be a triangle with $BC=30$, $AC=50$, and $AB=60$. Circle $\omega_B$ is the circle passing through $A$ and $B$ tangent to $BC$ at $B$; $\omega_C$ is defined similarly. Suppose the tangent to $\odot(ABC)$ at $A$ intersects $\omega_B$ and $\omega_C$ for the second time at $X$ and $Y$ respectively. Compute $XY$.
2018 CMIMC Team 3.1
Let $\Omega$ be a semicircle with endpoints $A$ and $B$ and diameter 3. Points $X$ and $Y$ are located on the boundary of $\Omega$ such that the distance from $X$ to $AB$ is $\frac{5}{4}$ and the distance from $Y$ to $AB$ is $\frac{1}{4}$. Compute \[(AX+BX)^2 - (AY+BY)^2.\]
2018 CMIMC Team 4.2Let $\Omega$ be a semicircle with endpoints $A$ and $B$ and diameter 3. Points $X$ and $Y$ are located on the boundary of $\Omega$ such that the distance from $X$ to $AB$ is $\frac{5}{4}$ and the distance from $Y$ to $AB$ is $\frac{1}{4}$. Compute \[(AX+BX)^2 - (AY+BY)^2.\]
Let $T = TNYWR$. Let $CMU$ be a triangle with $CM=13$, $MU=14$, and $UC=15$. Rectangle $WEAN$ is inscribed in $\triangle CMU$ with points $W$ and $E$ on $\overline{MU}$, point $A$ on $\overline{CU}$, and point $N$ on $\overline{CM}$. If the area of $WEAN$ is $T$, what is its perimeter?
2018 CMIMC Team 7.1
Let $ABCD$ be a unit square, and suppose that $E$ and $F$ are on $\overline{AD}$ and $\overline{AB}$ such that $AE = AF = \tfrac23$. Let $\overline{CE}$ and $\overline{DF}$ intersect at $G$. If the area of $\triangle CFG$ can be expressed as simplified fraction $\frac{p}{q}$, find $p + q$.
2018 CMIMC Team 8.1
Let $\triangle ABC$ be a triangle with $AB=3$ and $AC=5$. Select points $D, E,$ and $F$ on $\overline{BC}$ in that order such that $\overline{AD}\perp \overline{BC}$, $\angle BAE=\angle CAE$, and $\overline{BF}=\overline{CF}$. If $E$ is the midpoint of segment $\overline{DF}$, what is $BC^2$?
2018 CMIMC Team 10.2
Let $T = TNYWR$. Circles $\omega_1$ and $\omega_2$ intersect at $P$ and $Q$. The common external tangent $\ell$ to the two circles closer to $Q$ touches $\omega_1$ and $\omega_2$ at $A$ and $B$ respectively. Line $AQ$ intersects $\omega_2$ at $X$ while $BQ$ intersects $\omega_1$ again at $Y$. Let $M$ and $N$ denote the midpoints of $\overline{AY}$ and $\overline{BX}$, also respectively. If $AQ=\sqrt{T}$, $BQ=7$, and $AB=8$, then find the length of $MN$.
Suppose $X, Y, Z$ are collinear points in that order such that $XY = 1$ and $YZ = 3$. Let $W$ be a point such that $YW = 5$, and define $O_1$ and $O_2$ as the circumcenters of triangles $\triangle WXY$ and $\triangle WYZ$, respectively. What is the minimum possible length of segment $\overline{O_1O_2}$?
2019 CMIMC Team 3
Points $A(0,0)$ and $B(1,1)$ are located on the parabola $y=x^2$. A third point $C$ is positioned on this parabola between $A$ and $B$ such that $AC=CB=r$. What is $r^2$?
2019 CMIMC Team 4
Let $ABCD$ be a unit square, and suppose that $E$ and $F$ are on $\overline{AD}$ and $\overline{AB}$ such that $AE = AF = \tfrac23$. Let $\overline{CE}$ and $\overline{DF}$ intersect at $G$. If the area of $\triangle CFG$ can be expressed as simplified fraction $\frac{p}{q}$, find $p + q$.
Let $\triangle ABC$ be a triangle with $AB=3$ and $AC=5$. Select points $D, E,$ and $F$ on $\overline{BC}$ in that order such that $\overline{AD}\perp \overline{BC}$, $\angle BAE=\angle CAE$, and $\overline{BF}=\overline{CF}$. If $E$ is the midpoint of segment $\overline{DF}$, what is $BC^2$?
Let $T = TNYWR$. Circles $\omega_1$ and $\omega_2$ intersect at $P$ and $Q$. The common external tangent $\ell$ to the two circles closer to $Q$ touches $\omega_1$ and $\omega_2$ at $A$ and $B$ respectively. Line $AQ$ intersects $\omega_2$ at $X$ while $BQ$ intersects $\omega_1$ again at $Y$. Let $M$ and $N$ denote the midpoints of $\overline{AY}$ and $\overline{BX}$, also respectively. If $AQ=\sqrt{T}$, $BQ=7$, and $AB=8$, then find the length of $MN$.
The figure below depicts two congruent triangles with angle measures $40^\circ$, $50^\circ$, and $90^\circ$. What is the measure of the obtuse angle $\alpha$ formed by the hypotenuses of these two triangles?
Let $ABC$ be an equilateral triangle with side length $2$, and let $M$ be the midpoint of $\overline{BC}$. Points $X$ and $Y$ are placed on $AB$ and $AC$ respectively such that $\triangle XMY$ is an isosceles right triangle with a right angle at $M$. What is the length of $\overline{XY}$?
Suppose $\mathcal{T}=A_0A_1A_2A_3$ is a tetrahedron with $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$, $A_0A_1=5, A_0A_2=12$ and $A_0A_3=9$. A cube $A_0B_0C_0D_0E_0F_0G_0H_0$ with side length $s$ is inscribed inside $\mathcal{T}$ with $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$, and $G_0\in \triangle A_1A_2A_3$; what is $s$?
Let $MATH$ be a trapezoid with $MA=AT=TH=5$ and $MH=11$. Point $S$ is the orthocenter of $\triangle ATH$. Compute the area of quadrilateral $MASH$.
Let $ABC$ be a triangle with $AB=209$, $AC=243$, and $\angle BAC = 60^\circ$, and denote by $N$ the midpoint of the major arc $\widehat{BAC}$ of circle $\odot(ABC)$. Suppose the parallel to $AB$ through $N$ intersects $\overline{BC}$ at a point $X$. Compute the ratio $\tfrac{BX}{XC}$
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Denote by $\omega$ its incircle. A line $\ell$ tangent to $\omega$ intersects $\overline{AB}$ and $\overline{AC}$ at $X$ and $Y$ respectively. Suppose $XY=5$. Compute the positive difference between the lengths of $\overline{AX}$ and $\overline{AY}$.
Consider the following three lines in the Cartesian plane:$$\begin{cases}
\ell_1: & 2x - y = 7\\
\ell_2: & 5x + y = 42\\
\ell_3: & x + y = 14
\end{cases}$$and let $f_i(P)$ correspond to the reflection of the point $P$ across $\ell_i$. Suppose $X$ and $Y$ are points on the $x$ and $y$ axes, respectively, such that $f_1(f_2(f_3(X)))= Y$. Let $t$ be the length of segment $XY$; what is the sum of all possible values of $t^2$?
Let $ABCD$ be a square of side length $1$, and let $P_1, P_2$ and $P_3$ be points on the perimeter such that $\angle P_1P_2P_3 = 90^\circ$ and $P_1, P_2, P_3$ lie on different sides of the square. As these points vary, the locus of the circumcenter of $\triangle P_1P_2P_3$ is a region $\mathcal{R}$; what is the area of $\mathcal{R}$?
Suppose $ABC$ is a triangle, and define $B_1$ and $C_1$ such that $\triangle AB_1C$ and $\triangle AC_1B$ are isosceles right triangles on the exterior of $\triangle ABC$ with right angles at $B_1$ and $C_1$, respectively. Let $M$ be the midpoint of $\overline{B_1C_1}$; if $B_1C_1 = 12$, $BM = 7$ and $CM = 11$, what is the area of $\triangle ABC$?
Points $A(0,0)$ and $B(1,1)$ are located on the parabola $y=x^2$. A third point $C$ is positioned on this parabola between $A$ and $B$ such that $AC=CB=r$. What is $r^2$?
2019 CMIMC Team 4
Let $\triangle A_1B_1C_1$ be an equilateral triangle of area $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows. First, she flips a coin. If it comes up heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$. If it comes up tails, she instead constructs $A_2$ such that $C_1$ is the midpoint of $\overline{A_2B_1}$. She performs analogous operations on $B_2$ and $C_2$. What is the expected value of the area of $\triangle A_2B_2C_2$?
2019 CMIMC Team 10
Let $\triangle ABC$ be a triangle with side lengths $a$, $b$, and $c$. Circle $\omega_A$ is the $A$-excircle of $\triangle ABC$, defined as the circle tangent to $BC$ and to the extensions of $AB$ and $AC$ past $B$ and $C$ respectively. Let $\mathcal{T}_A$ denote the triangle whose vertices are these three tangency points; denote $\mathcal{T}_B$ and $\mathcal{T}_C$ similarly. Suppose the areas of $\mathcal{T}_A$, $\mathcal{T}_B$, and $\mathcal{T}_C$ are $4$, $5$, and $6$ respectively. Find the ratio $a:b:c$.
Let $\triangle ABC$ be a triangle with side lengths $a$, $b$, and $c$. Circle $\omega_A$ is the $A$-excircle of $\triangle ABC$, defined as the circle tangent to $BC$ and to the extensions of $AB$ and $AC$ past $B$ and $C$ respectively. Let $\mathcal{T}_A$ denote the triangle whose vertices are these three tangency points; denote $\mathcal{T}_B$ and $\mathcal{T}_C$ similarly. Suppose the areas of $\mathcal{T}_A$, $\mathcal{T}_B$, and $\mathcal{T}_C$ are $4$, $5$, and $6$ respectively. Find the ratio $a:b:c$.
2019 CMIMC Team 13
Points $A$, $B$, and $C$ lie in the plane such that $AB=13$, $BC=14$, and $CA=15$. A peculiar laser is fired from $A$ perpendicular to $\overline{BC}$. After bouncing off $BC$, it travels in a direction perpendicular to $CA$. When it hits $CA$, it travels in a direction perpendicular to $AB$, and after hitting $AB$ its new direction is perpendicular to $BC$ again. If this process is continued indefinitely, the laser path will eventually approach some finite polygonal shape $T_\infty$. What is the ratio of the perimeter of $T_\infty$ to the perimeter of $\triangle ABC$?
For every positive integer $k$, let $\mathbf{T}_k = (k(k+1), 0)$, and define $\mathcal{H}_k$ as the homothety centered at $\mathbf{T}_k$ with ratio $\tfrac{1}{2}$ if $k$ is odd and $\tfrac{2}{3}$ is $k$ is even. Suppose $P = (x,y)$ is a point such that $(\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20).$ What is $x+y$?
(A homothety $\mathcal{H}$ with nonzero ratio $r$ centered at a point $P$ maps each point $X$ to the point $Y$ on ray $\overrightarrow{PX}$ such that $PY = rPX$.)
Let $\mathcal E$ be an ellipse with foci $F_1$ and $F_2$. Parabola $\mathcal P$, having vertex $F_1$ and focus $F_2$, intersects $\mathcal E$ at two points $X$ and $Y$. Suppose the tangents to $\mathcal E$ at $X$ and $Y$ intersect on the directrix of $\mathcal P$. Compute the eccentricity of $\mathcal E$.
(A parabola $\mathcal P$ is the set of points which are equidistant from a point, called the focus of $\mathcal P$, and a line, called the directrix of $\mathcal P$. An ellipse $\mathcal E$ is the set of points $P$ such that the sum $PF_1 + PF_2$ is some constant $d$, where $F_1$ and $F_2$ are the foci of $\mathcal E$. The eccentricity of $\mathcal E$ is defined to be the ratio $F_1F_2/d$.)
Points $P$ and $Q$ lie on a circle $\omega$. The tangents to $\omega$ at $P$ and $Q$ intersect at point $T$, and point $R$ is chosen on $\omega$ so that $T$ and $R$ lie on opposite sides of $PQ$ and $\angle PQR = \angle PTQ$. Let $RT$ meet $\omega$ for the second time at point $S$. Given that $PQ = 12$ and $TR = 28$, determine $PS$.
Points $A$, $B$, and $C$ lie in the plane such that $AB=13$, $BC=14$, and $CA=15$. A peculiar laser is fired from $A$ perpendicular to $\overline{BC}$. After bouncing off $BC$, it travels in a direction perpendicular to $CA$. When it hits $CA$, it travels in a direction perpendicular to $AB$, and after hitting $AB$ its new direction is perpendicular to $BC$ again. If this process is continued indefinitely, the laser path will eventually approach some finite polygonal shape $T_\infty$. What is the ratio of the perimeter of $T_\infty$ to the perimeter of $\triangle ABC$?
Let $PQRS$ be a square with side length 12. Point $A$ lies on segment $\overline{QR}$ with $\angle QPA = 30^\circ$, and point $B$ lies on segment $\overline{PQ}$ with $\angle SRB = 60^\circ$. What is $AB$?
Let $ABC$ be a triangle. Points $D$ and $E$ are placed on $\overline{AC}$ in the order $A$, $D$, $E$, and $C$, and point $F$ lies on $\overline{AB}$ with $EF\parallel BC$. Line segments $\overline{BD}$ and $\overline{EF}$ meet at $X$. If $AD = 1$, $DE = 3$, $EC = 5$, and $EF = 4$, compute $FX$.
Point $A$, $B$, $C$, and $D$ form a rectangle in that order. Point $X$ lies on $CD$, and segments $\overline{BX}$ and $\overline{AC}$ intersect at $P$. If the area of triangle $BCP$ is 3 and the area of triangle $PXC$ is 2, what is the area of the entire rectangle?
Triangle $ABC$ has a right angle at $B$. The perpendicular bisector of $\overline{AC}$ meets segment $\overline{BC}$ at $D$, while the perpendicular bisector of segment $\overline{AD}$ meets $\overline{AB}$ at $E$. Suppose $CE$ bisects acute $\angle ACB$. What is the measure of angle $ACB$?
(A homothety $\mathcal{H}$ with nonzero ratio $r$ centered at a point $P$ maps each point $X$ to the point $Y$ on ray $\overrightarrow{PX}$ such that $PY = rPX$.)
Two circles $\omega_A$ and $\omega_B$ have centers at points $A$ and $B$ respectively and intersect at points $P$ and $Q$ in such a way that $A$, $B$, $P$, and $Q$ all lie on a common circle $\omega$. The tangent to $\omega$ at $P$ intersects $\omega_A$ and $\omega_B$ again at points $X$ and $Y$ respectively. Suppose $AB = 17$ and $XY = 20$. Compute the sum of the radii of $\omega_A$ and $\omega_B$.
In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $ABC$. The circumcircles of $BFI$ and $CEI$ intersect at $X \neq I$. Find the length of $DX$.
(A parabola $\mathcal P$ is the set of points which are equidistant from a point, called the focus of $\mathcal P$, and a line, called the directrix of $\mathcal P$. An ellipse $\mathcal E$ is the set of points $P$ such that the sum $PF_1 + PF_2$ is some constant $d$, where $F_1$ and $F_2$ are the foci of $\mathcal E$. The eccentricity of $\mathcal E$ is defined to be the ratio $F_1F_2/d$.)
In triangle $ABC$, points $M$ and $N$ are on segments $AB$ and $AC$ respectively such that $AM = MC$ and $AN = NB$. Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $ABC$. Given that the perimeters of $PMN$ and $BCNM$ are $21$ and $29$ respectively, and that $PB = 5$, compute the length of $BC$.
Four copies of an acute scalene triangle $\mathcal T$, one of whose sides has length $3$, are joined to form a tetrahedron with volume $4$ and surface area $24$. Compute the largest possible value for the circumradius of $\mathcal T$.
Let $ABC$ be a triangle with centroid $G$ and $BC = 3$. If $ABC$ is similar to $GAB$, compute the area of $ABC$.
2020 CMIMC Team 10
Let $ABC$ be a triangle. The incircle $\omega$ of $\triangle ABC$, which has radius $3$, is tangent to $\overline{BC}$ at $D$. Suppose the length of the altitude from $A$ to $\overline{BC}$ is $15$ and $BD^2 + CD^2 = 33$. What is $BC$?
Let $ABC$ be a triangle. The incircle $\omega$ of $\triangle ABC$, which has radius $3$, is tangent to $\overline{BC}$ at $D$. Suppose the length of the altitude from $A$ to $\overline{BC}$ is $15$ and $BD^2 + CD^2 = 33$. What is $BC$?
2020 CMIMC Team 15
Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose $M$ is the midpoint of segment $\overline{BC}$, $N$ is the midpoint of $\overline{AM}$, and $E$ and $F$ are the feet of the altitudes of $M$ onto $\overline{AB}$ and $\overline{AC}$, respectively. Further suppose $BC$ intersects $NE$ at $S$ and $NF$ at $T$, and let $X$ and $Y$ be the circumcenters of $\triangle MES$ and $\triangle MFT$, respectively. If $XY$ is tangent to the circumcircle of $\triangle ABC$, what is the area of $\triangle ABC$?
Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose $M$ is the midpoint of segment $\overline{BC}$, $N$ is the midpoint of $\overline{AM}$, and $E$ and $F$ are the feet of the altitudes of $M$ onto $\overline{AB}$ and $\overline{AC}$, respectively. Further suppose $BC$ intersects $NE$ at $S$ and $NF$ at $T$, and let $X$ and $Y$ be the circumcenters of $\triangle MES$ and $\triangle MFT$, respectively. If $XY$ is tangent to the circumcircle of $\triangle ABC$, what is the area of $\triangle ABC$?
Triangle $ABC$ has a right angle at $A$, $AB=20$, and $AC=21$. Circles $\omega_A$, $\omega_B$, and $\omega_C$ are centered at $A$, $B$, and $C$ respectively and pass through the midpoint $M$ of $\overline{BC}$. $\omega_A$ and $\omega_B$ intersect at $X\neq M$, and $\omega_A$ and $\omega_C$ intersect at $Y\neq M$. Find $XY$.
by Connor Gordon
Points $A$, $B$, and $C$ lie on a line, in that order, with $AB=8$ and $BC=2$. $B$ is rotated $20^\circ$ counter-clockwise about $A$ to a point $B'$, tracing out an arc $R_1$. $C$ is then rotated $20^\circ$ clockwise about $A$ to a point $C'$, tracing out an arc $R_2$. What is the area of the region bounded by arc $R_1$, segment $B'C$, arc $R_2$, and segment $C'B$?
by Thomas Lam
Consider trapezoid $[ABCD]$ which has $AB\parallel CD$ with $AB = 5$ and $CD = 9$. Moreover, $\angle C = 15^\circ$ and $\angle D = 75^\circ$. Let $M_1$ be the midpoint of $AB$ and $M_2$ be the midpoint of $CD$. What is the distance $M_1M_2$?
by Daniel Li
A $2\sqrt5$ by $4\sqrt5$ rectangle is rotated by an angle $\theta$ about one of its diagonals. If the total volume swept out by the rotating rectangle is $62\pi$, find the measure of $\theta$ in degrees.
by Connor Gordon
Emily is at $(0,0)$, chilling, when she sees a spider located at $(1,0)$! Emily runs a continuous path to her home, located at $(\sqrt{2}+2,0)$, such that she is always moving away from the spider and toward her home. That is, her distance from the spider always increases whereas her distance to her home always decreases. What is the area of the set of all points that Emily could have visited on her run home?
by Thomas Lam
In convex quadrilateral $ABCD$, $\angle ADC = 90^\circ + \angle BAC$. Given that $AB = BC = 17$, and $CD = 16$, what is the maximum possible area of the quadrilateral?
by Thomas Lam
Let $\triangle ABC$ be a triangle with $AB=10$ and $AC=16,$ and let $I$ be the intersection of the internal angle bisectors of $\triangle ABC.$ Suppose the tangents to the circumcircle of $\triangle BIC$ at $B$ and $C$ intersect at a point $P$ with $PA=8.$ Compute the length of ${BC}.$
by Kyle Lee
Let $ABCDEF$ be an equilateral heaxagon such that $\triangle ACE \cong \triangle DFB$. Given that $AC = 7$, $CE=8$, and $EA=9$, what is the side length of this hexagon?
by Thomas Lam
Let $\gamma_1, \gamma_2, \gamma_3$ be three circles with radii $3, 4, 9,$ respectively, such that $\gamma_1$ and $\gamma_2$ are externally tangent at $C,$ and $\gamma_3$ is internally tangent to $\gamma_1$ and $\gamma_2$ at $A$ and $B,$ respectively. Suppose the tangents to $\gamma_3$ at $A$ and $B$ intersect at $X.$ The line through $X$ and $C$ intersect $\gamma_3$ at two points, $P$ and $Q.$ Compute the length of $PQ.$
by Kyle Lee
Let circles $\omega$ and $\Gamma$, centered at $O_1$ and $O_2$ and having radii $42$ and $54$ respectively, intersect at points $X,Y$, such that $\angle O_1XO_2 = 105^{\circ}$. Points $A$, $B$ lie on $\omega$ and $\Gamma$ respectively such that $\angle O_1XA = \angle AXB = \angle BXO_2$ and $Y$ lies on both minor arcs $XA$ and $XB$. Define $P$ to be the intersection of $AO_2$ and $BO_1$. Suppose $XP$ intersects $AB$ at $C$. What is the value of $\frac{AC}{BC}$?
by Puhua Cheng
Convex pentagon $ABCDE$ has $\overline{BC}=17$, $\overline{AB}=2\overline{CD}$, and $\angle E=90^\circ$. Additionally, $\overline{BD}-\overline{CD}=\overline{AC}$, and $\overline{BD}+\overline{CD}=25$. Let $M$ and $N$ be the midpoints of $BC$ and $AD$ respectively. Ray $EA$ is extended out to point $P$, and a line parallel to $AD$ is drawn through $P$, intersecting line $EM$ at $Q$. Let $G$ be the midpoint of $AQ$. Given that $N$ and $G$ lie on $EM$ and $PM$ respectively, and the perimeter of $\triangle QBC$ is $42$, find the length of $\overline{EM}$.
by Adam Bertelli
Let $ABC$ be a triangle with $AB < AC$ and $\omega$ be a circle through $A$ tangent to both the $B$-excircle and the $C$-excircle. Let $\omega$ intersect lines $AB, AC$ at $X,Y$ respectively and $X,Y$ lie outside of segments $AB, AC$. Let $O$ be the center of $\omega$ and let $OI_C, OI_B$ intersect line $BC$ at $J,K$ respectively. Suppose $KJ = 4$, $KO = 16$ and $OJ = 13$. Find $\frac{[KI_BI_C]}{[JI_BI_C]}$.
by Grant Yu
Given a trapezoid with bases $AB$ and $CD$, there exists a point $E$ on $CD$ such that drawing the segments $AE$ and $BE$ partitions the trapezoid into $3$ similar isosceles triangles, each with long side twice the short side. What is the sum of all possible values of $\frac{CD}{AB}$?
by Adam Bertelli
Let $P$ and $Q$ be fixed points in the Euclidean plane. Consider another point $O_0$. Define $O_{i+1}$ as the center of the unique circle passing through $O_i$, $P$ and $Q$. (Assume that $O_i,P,Q$ are never collinear.) How many possible positions of $O_0$ satisfy that $O_{2021}=O_{0}$?
by Fei Peng
Let $ABC$ be a triangle with circumcenter $O$. Additionally, $\angle BAC=20^\circ$ and $\angle BCA = 70^\circ$. Let $D, E$ be points on side $AC$ such that $BO$ bisects $\angle ABD$ and $BE$ bisects $\angle CBD$. If $P$ and $Q$ are points on line $BC$ such that $DP$ and $EQ$ are perpendicular to $AC$, what is $\angle PAQ$?
by Daniel Li
Let $\triangle ABC$ be a triangle, and let $l$ be the line passing through its incenter and centroid. Assume that $B$ and $C$ lie on the same side of $l$, and that the distance from $B$ to $l$ is twice the distance from $C$ to $l$. Suppose also that the length $BA$ is twice that of $CA$. If $\triangle ABC$ has integer side lengths and is as small as possible, what is $AB^2+BC^2+CA^2$?
by Thomas Lam
Let $S$ be the set of lattice points $(x,y) \in \mathbb{Z}^2$ such that $-10\leq x,y \leq 10$. Let the point $(0,0)$ be $O$. Let Scotty the Dog's position be point $P$, where initially $P=(0,1)$. At every second, consider all pairs of points $C,D \in S$ such that neither $C$ nor $D$ lies on line $OP$, and the area of quadrilateral $OCPD$ (with the points going clockwise in that order) is $1$. Scotty finds the pair $C,D$ maximizing the sum of the $y$ coordinates of $C$ and $D$, and randomly jumps to one of them, setting that as the new point $P$. After $50$ such moves, Scotty ends up at point $(1, 1)$. Find the probability that he never returned to the point $(0,1)$ during these $50$ moves.
by David Tang
Adam has a circle of radius $1$ centered at the origin.
- First, he draws $6$ segments from the origin to the boundary of the circle, which splits the upper (positive $y$) semicircle into $7$ equal pieces.
- Next, starting from each point where a segment hit the circle, he draws an altitude to the $x$-axis.
- Finally, starting from each point where an altitude hit the $x$-axis, he draws a segment directly away from the bottommost point of the circle $(0,-1)$, stopping when he reaches the boundary of the circle.
What is the product of the lengths of all $18$ segments Adam drew?
by Adam Bertelli
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