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: 42

2017 ASDAN Math Tournament, 9
Tags: geometry test
Triangle $ABC$ is isosceles with $AC=BC=25$ and $AB=10$. Let $O$ be the orthocenter of $\triangle ABC$, the intersection of the three altitudes of $\triangle ABC$. Reflect $O$ across $AB$ to a point $D$, and extend $CB$ and $AD$ to intersect at point $E$. Compute the area of $\triangle ABE$.

2016 ASDAN Math Tournament, 9
Tags: geometry test
A cyclic quadrilateral $ABCD$ has side lengths $AB=14$, $BC=19$, $CD=26$, and $DA=29$. Compute the sine of the smaller angle between diagonals $AC$ and $BD$.

2017 ASDAN Math Tournament, 2
Tags: geometry test
An equilateral triangle $ABC$ shares a side with a square $BCDE$. If the resulting pentagon has a perimeter of $20$, what is the area of the pentagon? (The triangle and square do not overlap).

2014 ASDAN Math Tournament, 5
Tags: geometry test
Consider a triangle $ABC$ with $AB=4$, $BC=3$, and $AC=2$. Let $D$ be the midpoint of line $BC$. Find the length of $AD$.

2017 ASDAN Math Tournament, 4
Tags: geometry test
An ant starts at corner $A$ of a square room $ABCD$ with side length $2\sqrt{2}$. In the middle of the room, there is a circular pillar of radius $1$ centered at the center of $ABCD$. What is the minimum distance it has to travel to get to corner $C$?

2016 ASDAN Math Tournament, 1
Tags: geometry test
Compute the area of the trapezoid $ABCD$ with right angles $BAD$ and $ADC$ and side lengths of $AB=3$, $BC=5$, and $CD=7$.

2015 ASDAN Math Tournament, 10
Tags: geometry test
Triangle $ABC$ has $\angle BAC=90^\circ$. A semicircle with diameter $XY$ is inscribed inside $\triangle ABC$ such that it is tangent to a point $D$ on side $BC$, with $X$ on $AB$ and $Y$ on $AC$. Let $O$ be the midpoint of $XY$. Given that $AB=3$, $AC=4$, and $AX=\tfrac{9}{4}$, compute the length of $AO$.

2016 ASDAN Math Tournament, 2
Tags: geometry test
Two concentric circles have differing radii such that a chord of the outer circle which is tangent to the inner circle has length $18$. Compute the area inside the bigger circle which lies outside of the smaller circle.

2014 ASDAN Math Tournament, 2
Tags: geometry test
Let $ABC$ be a triangle with sides $AB=19$, $BC=21$, and $AC=20$. Let $\omega$ be the incircle of $ABC$ with center $I$. Extend $BI$ so that it intersects $AC$ at $E$. If $\omega$ is tangent to $AC$ at the point $D$, then compute the length of $DE$.

2016 ASDAN Math Tournament, 5
Tags: geometry test
Let $\Gamma_1$ be a circle of radius $6$, and let $\Gamma_2$ be a circle of radius $1$. Next, let the circles be internally tangent at point $P$, and let $AP$ be a diameter of circle $\Gamma_1$. Finally, let $Y$ be a point on $\Gamma_2$ such that $AY$ is tangent to it. Compute the length of $PY$.

2016 ASDAN Math Tournament, 3
Tags: geometry test
Let $ABCD$ be a unit square, and let there be two unit circles centered at $C$ and $D$. Let $P$ be the point of intersection of the two circles inside the square. Compute $\angle APB$ in degrees.

2017 ASDAN Math Tournament, 7
Tags: geometry test
Three identical circles are packed into a unit square. Each of the three circles are tangent to each other and tangent to at least one side of the square. If $r$ is the maximum possible radius of the circle, what is $(2-\tfrac{1}{r})^2$?

2016 ASDAN Math Tournament, 8
Tags: geometry test
A circle with center $O$ is drawn in the first quadrant of the 2D Cartesian plane (the quadrant with both positive $x$ and $y$ values) such that it lies tangent to the $x$ and $y$-axes. A line is drawn with slope $m>1$ and passing through the origin; the line intersects the circle at two points $A$ and $B$, with $A$ closer to the origin than $B$. Suppose that $ABO$ is an equilateral triangle. Compute $m$.

2014 ASDAN Math Tournament, 10
Tags: geometry test
In a convex quadrilateral $ABCD$ we are given that $\angle CAD=10^\circ$, $\angle DBC=20^\circ$, $\angle BAD=40^\circ$, $\angle ABC=50^\circ$. Compute angle $BDC$.

2016 ASDAN Math Tournament, 4
Tags: geometry test
Let $ABCD$ be a rectangle with $AB=9$ and $BC=3$. Suppose that $D$ is reflected across $AC$ to a point $E$. Compute the area of trapezoid $AEBC$.

2017 ASDAN Math Tournament, 8
Tags: geometry test
Let $\triangle ABC$ be a right triangle with right angle $\angle ACB$. Square $DEFG$ is contained inside triangle $ABC$ such that $D$ lies on $AB$, $E$ lies on $BC$, $F$ lies on $AC$, $AD=AF$, and $GA=GD=GF$. Suppose that $CE=2$. If $M$ is the area of triangle $ABC$ and $N$ is the area of square $DEFG$, compute $M-N$.

2016 ASDAN Math Tournament, 6
Tags: geometry test
In the diagram below, square $ABCD$ has side length $4$. Two congruent square $EGIK$ and $FHJL$ are drawn such that $AE=FB=BG=HC=CI=JD=DK=LA=1$ and $EF=GH=IJ=KL=2$. Compute the area of the region that lies in both $EGIK$ and $FHJL$.

2015 ASDAN Math Tournament, 2
Tags: geometry test
There is a rectangular field that measures $20\text{m}$ by $15\text{m}$. Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\text{m}$ sides such that he is $6\text{m}$ from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two parts with equal area. How far in meters did Xiaoyu fly?

2017 ASDAN Math Tournament, 1
Tags: geometry test
What is the surface area of a cube with volume $64$?

2015 ASDAN Math Tournament, 7
Tags: geometry test
In a rectangle $ABCD$, two segments $EG$ and $FH$ divide it into four smaller rectangles. $BH$ intersects $EG$ at $X$, $CX$ intersects $HF$ and $Y$, $DY$ intersects $EG$ at $Z$. Given that $AH=4$, $HD=6$, $AE=4$, and $EB=5$, find the area of quadrilateral $HXYZ$.

2017 ASDAN Math Tournament, 10
Tags: geometry test
Triangle $ABC$ is inscribed in circle $\gamma_1$ with radius $r_1$. Let $\gamma_2$ (with radius $r_2$) be the circle internally tangent to $\gamma_1$ at $A$ and tangent to $BC$ at $D$. Let $I$ be the incenter of $ABC$, and $P$ and $Q$ be the intersection of $\gamma_2$ with $AB$ and $AC$ respectively. Given that $P$, $I$, and $Q$ are collinear, $AI=25$, and the circumradius of triangle $BIC$ is $24$, compute the ratio of the radii $\tfrac{r_2}{r_1}$.

2014 ASDAN Math Tournament, 1
Tags: geometry test
Consider a square of side length $1$ and erect equilateral triangles of side length $1$ on all four sides of the square such that one triangle lies inside the square and the remaining three lie outside. Going clockwise around the square, let $A$, $B$, $C$, $D$ be the circumcenters of the four equilateral triangles. Compute the area of $ABCD$.

2014 ASDAN Math Tournament, 7
Tags: geometry test
Let $ABCD$ be a square piece of paper with side length $4$. Let $E$ be a point on $AB$ such that $AE=3$ and let $F$ be a point on $CD$ such that $DF=1$. Now, fold $AEFD$ over the line $EF$. Compute the area of the resulting shape.

2015 ASDAN Math Tournament, 4
Tags: geometry test
In trapezoid $ABCD$ with $AD\parallel BC$, $AB=6$, $AD=9$, and $BD=12$. If $\angle ABD=\angle DCB$, find the perimeter of the trapezoid.

2015 ASDAN Math Tournament, 5
Tags: geometry test
The eight corners of a cube are cut off, yielding a polyhedron with $6$ octagonal faces and $8$ triangular faces. Given that all polyhedron's edges have length $2$, compute the volume of the polyhedron.