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.

AND:
OR:
NO:

Found problems: 42

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

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, 9
Tags: geometry test
Regular tetrahedron $ABCD$ has center $O$ and side length $1$. Points $A'$, $B'$, $C'$, and $D'$ are defined by reflecting $A$, $B$, $C$, and $D$ about $O$. Compute the volume of the polyhedron with vertices $ABCDA'B'C'D'$.

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 $PY$.

2015 ASDAN Math Tournament, 8
Tags: geometry test
In triangle $ABC$, point $D$ is on side $BC$ such that $AD$ is the angle bisector of $\angle BAC$. If $AB=12$, $AD=9$, and $AC=15$, compute $\cos\tfrac{\angle BAC}{2}$.

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, 6
Tags: geometry test
Consider a circle of radius $4$ with center $O_1$, a circle of radius $2$ with center $O_2$ that lies on the circumference of circle $O_1$, and a circle of radius $1$ with center $O_3$ that lies on the circumference of circle $O_2$. The centers of the circle are collinear in the order $O_1$, $O_2$, $O_3$. Let $A$ be a point of intersection of circles $O_1$ and $O_2$ and $B$ be a point of intersection of circles $O_2$ and $O_3$ such that $A$ and $B$ lie on the same semicircle of $O_2$. Compute the length of $AB$.

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

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, 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.

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 $(2-\tfrac{1}{r})^2$?

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 $\tfrac{r_2}{r_1}$.

2017 ASDAN Math Tournament, 5
Tags: geometry test
Regular hexagon $ABCDEF$ has side length $2$. Line segment $BD$ is drawn, and circle $O$ is inscribed inside the pentagon $ABDEF$ such that $O$ is tangent to $AF$, $BD$, and $EF$. Compute the radius of $O$.

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

2015 ASDAN Math Tournament, 1
Tags: geometry test
Four unit circles are placed on a square of side length $2$, with each circle centered on one of the four corners of the square. Compute the area of the square which is not contained within any of the four circles.

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 $HXYZ$.

2014 ASDAN Math Tournament, 4
Tags: geometry test
Consider a square $ABCD$ with side length $4$ and label the midpoint of side $BC$ as $M$. Let $X$ be the point along $AM$ obtained by dropping a perpendicular from $D$ onto $AM$. Compute the product of the lengths $XC$ and $MD$.