This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 18

2014 ASDAN Math Tournament, 1
Tags: 2014 , Geometry Tiebreaker
Points $A$, $B$, $C$, and $D$ lie in the plane with $AB=AD=7$, $CB=CD=4$, and $BD=6$. Compute the sum of all possible values of $AC$.

2019 ASDAN Math Tournament, 3
Tags: Geometry Tiebreaker , 2019
Consider a triangle $\vartriangle ABC$ with $BC = 10$. An excircle is a circle that is tangent to one side of the triangle as well as the extensions of the other two sides; suppose that the excircle opposite vertex $B$ has center $I_2$ and exradius $r_2 = 11$, and suppose that the excircle opposite vertex $C$ has center $I_3$ and exradius $r_3 = 13$. Compute $I_2I_3$.

2019 ASDAN Math Tournament, 2
Tags: Geometry Tiebreaker , 2019
Consider a triangle $\vartriangle ABC$ with $AB = 5$ and $BC = 4$. Let $G$ be the centroid of the triangle, and let $P$ lie on line $AG$ such that $AG = GP$ and $P\ne A$. Suppose that $P$ lies on the circumcircle of $\vartriangle ABC$. Compute $CA$.

2014 ASDAN Math Tournament, 3
Tags: 2014 , Geometry Tiebreaker
Let $ABC$ be a triangle and $I$ its incenter. Suppose $AI=\sqrt{2}$, $BI=\sqrt{5}$, $CI=\sqrt{10}$ and the inradius is $1$. Let $A'$ be the reflection of $I$ across $BC$, $B'$ the reflection across $AC$, and $C'$ the reflection across $AB$. Compute the area of triangle $A'B'C'$.

2018 ASDAN Math Tournament, 2
Tags: Geometry Tiebreaker , 2018
In 3D coordinate space, $O$ is the origin, $A$ lies on the positive $x$-axis, $B$ lies on the positive $y$-axis, and $C$ lies on the positive $z$-axis such that $BO = 2AO$ and $CO = 3AO$. Suppose that a unit cube with sides parallel to the axes can be inscribed within tetrahedron $ABCO$. Compute $AO + BO + CO$.

2017 ASDAN Math Tournament, 3
Tags: 2017 , Geometry Tiebreaker
Triangle $ABC$ has $AB=4,BC=6,CA=5$. Let $M$ be the midpoint of $\overline{BC}$ and $P$ the point on the circumcircle of $\triangle ABC$ such that $\angle MPA=90^\circ$. Let points $D$ and $E$ lie on $\overline{AC}$ and $\overline{AB}$ respectively such that $\overline{BD}\perp\overline{AC}$ and $\overline{CE}\perp\overline{AB}$. Find $\tfrac{PD}{PE}$.

2015 ASDAN Math Tournament, 3
Tags: 2015 , Geometry Tiebreaker
Place points $A$, $B$, $C$, $D$, $E$, and $F$ evenly spaced on a unit circle. Compute the area of the shaded $12$-sided region, where the region is bounded by line segments $AD$, $DF$, $FB$, $BE$, $EC$, and $CA$. [center]<see attached>[/center]

2015 ASDAN Math Tournament, 2
Tags: 2015 , Geometry Tiebreaker
Let $ABCD$ be a square with side length $5$, and let $E$ be the midpoint of $CD$. Let $F$ be the point on $AE$ such that $CF=5$. Compute $AF$.

2014 ASDAN Math Tournament, 2
Tags: 2014 , Geometry Tiebreaker
Let $RICE$ be a quadrilateral with an inscribed circle $O$ such that every side of $RICE$ is tangent to $O$. Given taht $RI=3$, $CE=8$, and $ER=7$, compute $IC$.

2018 ASDAN Math Tournament, 1
Tags: Geometry Tiebreaker , 2018
Point $X$ is placed on segment $AB$ of a regular hexagon $ABCDEF$ such that the ratio of the area of $AXEF$ to the area of $XBCDE$ is $\frac12$. If $AB = 2018$, find $AX$.

2017 ASDAN Math Tournament, 1
Tags: 2017 , Geometry Tiebreaker
In $\triangle ABC$, we have $\angle ABC=20^\circ$. In addition, $D$ is drawn on $\overline{AB}$ such that $AC=CD=BD$. Compute $\angle ACD$ in degrees.

2016 ASDAN Math Tournament, 2
Tags: 2016 , Geometry Tiebreaker
A four-pointed star is formed by placing for equilateral triangles of side length $4$ in a coordinate grid. The triangles are placed such that their bases lie along one of the coordinate axes, with the midpoint of the bases lying at the origin, and such that the vertices opposite the bases lie at four distinct points. Compute the area contained within the star.

2016 ASDAN Math Tournament, 1
Tags: 2016 , Geometry Tiebreaker
$ABCDE$ is a pentagon where $AB=12$, $BC=20$, $CD=7$, $DE=24$, $EA=9$, and $\angle EAB=\angle CDE=90^\circ$. Compute the area of the pentagon.

2018 ASDAN Math Tournament, 3
Tags: Geometry Tiebreaker , 2018
In $\vartriangle ABC$, $AC > AB$. $B$ is reflected across $\overline{AC}$ to a point $D$, and $C$ is reflected across $\overline{AD}$ to a point $E$. Suppose that $AC = 6\sqrt3 + 6$, $BC = 6$, and $\overline{BC} \parallel \overline{AE}$. Compute $AB$.

2015 ASDAN Math Tournament, 1
Tags: 2015 , Geometry Tiebreaker
A rectangle $ABCD$ is split into four smaller non-overlapping rectangles by two perpendicular line segments, whose endpoints are on the sides of $ABCD$. If the smallest three rectangles have areas of $48$, $18$, and $12$, what is the area of $ABCD$?

2016 ASDAN Math Tournament, 3
Tags: 2016 , Geometry Tiebreaker
Let $H$ be the orthocenter of triangle $ABC$, and $D$ be the foot of $A$ onto $BC$. Given that $DB=3$, $DH=2$, and $DC=6$, calculate $HA$.

2017 ASDAN Math Tournament, 2
Tags: 2017 , Geometry Tiebreaker
Circles $A,B,C$ are externally tangent. Let $P$ be the tangent point between circles $A$ and $C$, and $Q$ be the tangent point between circles $B$ and $C$. Let $r_C$ be the radius of circle $C$. If the chord connecting $P$ and $Q$ has length $r_C\sqrt{2}$ and the radii of circles $A$ and $B$ are $4$ and $7$, respectively, what is the radius of circle $C$?

2019 ASDAN Math Tournament, 1
Tags: Geometry Tiebreaker , 2019
A kite is a quadrilateral with $2$ pairs of equal adjacent sides. Given a cyclic kite with side lengths $3$ and $4$, compute the distance between the intersection of its diagonals and the center of the circle circumscribing it.