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

2015 ASDAN Math Tournament, 6
Tags: 2015 , Geometry Test
Let $ABC$ be a triangle and let $D$ be a point on $AC$. The angle bisector of $\angle BAC$ intersects $BD$ at $E$ and $BC$ at $F$. Suppose that $\tfrac{CF}{DE}=\tfrac{5}{4}$ and that $\tfrac{BE}{BF}=\tfrac{3}{2}$. What is $\tfrac{CD}{AD}$?

2014 ASDAN Math Tournament, 7
Tags: 2014 , 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, 10
Tags: 2015 , 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 $AO$.

2015 ASDAN Math Tournament, 7
Tags: 2015 , 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 $HXYZ$.

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

2016 ASDAN Math Tournament, 5
Tags: 2016 , 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 $PY$.

2017 ASDAN Math Tournament, 8
Tags: 2017 , 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, 3
Tags: 2016 , 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.

2018 ASDAN Math Tournament, 3
Tags: Geometry Test , 2018
In parallelogram $ABCD$, $AB = 10$, and $AB = 2BC$. Let $M$ be the midpoint of $CD$, and suppose that $BM = 2AM$. Compute $AM$.

2017 ASDAN Math Tournament, 4
Tags: 2017 , 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, 10
Tags: 2016 , Geometry Test
Compute the radius of the sphere inscribed in the tetrahedron with coordinates $(2,0,0)$, $(4,0,0)$, $(0,1,0)$, and $(0,0,3)$.

2015 ASDAN Math Tournament, 4
Tags: 2015 , 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.

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

2017 ASDAN Math Tournament, 2
Tags: 2017 , 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).

2015 ASDAN Math Tournament, 5
Tags: 2015 , 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.

2016 ASDAN Math Tournament, 4
Tags: 2016 , 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, 6
Tags: 2017 , Geometry Test
Let $\triangle ABC$ be a right triangle with right angle $\angle B$. Suppose the angle bisector $l$ of $B$ divides the hypotenuse $AC$ into two segments of length $\sqrt{3}-1$ and $\sqrt{3}+1$. What is the measure of the smaller angle between $l$ and $AC$, in radians?