Found problems: 42
2015 ASDAN Math Tournament, 10
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 $AO$.
2014 ASDAN Math Tournament, 10
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
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 $AEBC$.
2015 ASDAN Math Tournament, 3
Points $E$ and $F$ are chosen on sides $BC$ and $CD$ respectively of rhombus $ABCD$ such that $AB=AE=AF=EF$, and $FC,DF,BE,EC>0$. Compute the measure of $\angle ABC$.
2017 ASDAN Math Tournament, 6
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?
2016 ASDAN Math Tournament, 8
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 $m$.
2016 ASDAN Math Tournament, 6
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 $FHJL$.
2018 ASDAN Math Tournament, 2
The intersection of $2$ cubes of side length $5$ is a cube of side length $3$. Compute the surface area of the entire figure.
2016 ASDAN Math Tournament, 2
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.
2017 ASDAN Math Tournament, 2
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).
2017 ASDAN Math Tournament, 8
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 $M-N$.
2014 ASDAN Math Tournament, 8
Moor made a lopsided ice cream cone. It turned out to be an oblique circular cone with the vertex directly above the perimeter of the base (see diagram below). The height and base radius are both of length $1$. Compute the radius of the largest spherical scoop of ice cream that it can hold such that at least $50\%$ of the scoop’s volume lies inside the cone.
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2014 ASDAN Math Tournament, 5
Consider a triangle $ABC$ with $AB=4$, $BC=3$, and $AC=2$. Let $D$ be the midpoint of line $BC$. Find the length of $AD$.
2014 ASDAN Math Tournament, 9
We have squares $ABCD$ and $EFGH$. Square $ABCD$ has points with coordinates $A=(1,1,-1)$, $B=(1,-1,-1)$, $C=(-1,-1,-1)$ and $D=(-1,1,-1)$. Square $EFGH$ has points with coordinates $E=(\sqrt{2},0,1)$, $F=(0,-\sqrt{2},1)$, $G=(-\sqrt{2},0,1)$, and $H=(0,\sqrt{2},1)$. Consider the solid formed by joining point $A$ to $H$ and $E$, point $B$ to $E$ and $F$, point $C$ to $F$ and $G$, and point $D$ to $G$ and $H$. Compute the volume of this solid.
2014 ASDAN Math Tournament, 2
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, 9
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, 9
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 $\triangle ABE$.
2017 ASDAN Math Tournament, 4
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 $C$?
2014 ASDAN Math Tournament, 3
Compute the perimeter of the triangle that has area $3-\sqrt{3}$ and angles $45^\circ$, $60^\circ$, and $75^\circ$.
2015 ASDAN Math Tournament, 2
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?
2016 ASDAN Math Tournament, 1
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, 6
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 $\tfrac{CD}{AD}$?
2016 ASDAN Math Tournament, 3
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.
2016 ASDAN Math Tournament, 7
The side lengths of triangle $ABC$ are $13$, $14$, and $15$. Let $I$ be the incenter of the triangle. Compute the product $AI\cdot BI\cdot CI$.
2017 ASDAN Math Tournament, 3
Line segment $AB$ has length $10$. A circle centered at $A$ has radius $5$, and a circle centered at $B$ has radius $5\sqrt{3}$. What is the area of the intersection of the two circles?