In triangle $ABC$, where $AC$ > $AB$, $M$ is the midpoint of $BC$ and $D$ is on segment $AC$ such that $DM$ is perpendicular to $BC$. Given that the areas of $MAD$ and $MBD$ are 5 and 6, respectively, compute the area of triangle $ABC$.
Equilateral triangles $ABE$, $BCF$, $CDG$ and $DAH$ are constructed outside the unit square $ABCD$. Eliza wants to stand inside octagon $AEBFCGDH$ so that she can see every point in the octagon without being blocked by an edge. What is the area of the region in which she can stand?
In triangle $ABC$, $AB = 2BC$. Given that $M$ is the midpoint of $AB$ and $\angle MCA = 60^{\circ}$, compute $\frac{CM}{AC}$ .
What is the maximum number of spheres with radius 1 that can fit into a sphere with radius 2?
Quadrilateral $ABCD$ satisfies $AB = 4$, $BC = 5$, $DA = 4$, $\angle DAB = 60^{\circ}$, and $\angle ABC = 150^{\circ}$. Find the area of $ABCD$.
The parabola $y = 2x^2$ is the wall of a fortress. Totoro is located at (0, 4) and fires a cannonball in a straight line at the closest point on the wall. Compute the $y$-coordinate of the point on the wall that the cannonball hits.
In rectangle $ABCD$, points $E$ and $F$ are on sides $AB$ and $CD$, respectively, such that $AE = CF > AD$ and $\angle CED = 90^{\circ}$. Lines $AF$, $BF$, $CE$ and $DE$ enclose a rectangle whose area is 24% of the area of $ABCD$. Compute $\frac{BF}{CE}$ .
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle BCD = 108^{\circ}$, $\angle CDE = 168^{\circ}$ and $AB =BC = CD = DE$. Find the measure of $\angle AEB$.
A semicircle with diameter $AB$ is constructed on the outside of rectangle $ABCD$ and has an arc length equal to the length of $BC$. Compute the ratio of the area of the rectangle to the area of the semicircle.
Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $BC$ = 10 and $AD$ = 18. Given that the two circles with diameters $BC$ and $AD$ are tangent, find the perimeter of $ABCD$.
Let $A$ be the answer to problem 13, and let $C$ be the answer to problem 15. In the interior of angle $NOM$ = $45^{\circ}$, there is a point $P$ such that $\angle MOP = A^{\circ}$ and $OP = C$. Let $X$ and $Y$ be the reflections of $P$ over $MO$ and $NO$, respectively. Find $(XY )^2$.
Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $AB$ = 4, $CD$ = 8, $BC$ = 5, and $AD$ = 6. Given that point $E$ is on segment $CD$ and that $AE$ is parallel to $BC$, find the ratio between the area of trapezoid $ABCD$ and the area of triangle $ABE$.
Let line segment $AB$ have length 25 and let points $C$ and $D$ lie on the same side of line $AB$ such that $AC$ = 15, $AD$ = 24, $BC$ = 20, and $BD$ = 7. Given that rays $AC$ and $BD$ intersect at point $E$, compute $EA + EB$.
The four points $(x, y)$ that satisfy $x = y^2 - 37$ and $y = x^2 -37$ form a convex quadrilateral in the coordinate plane. Given that the diagonals of this quadrilateral intersect at point $P$, find the coordinates of $P$ as an ordered pair.
Let $AB$ be the diameter of a semicircle. $C$, $D$, and $E$ are points on $AB$, in that order, such that $AC=1$, $CD=3$, $DE=4$, and $EB=2$. From $C$ and $E$, draw perpendicular lines of $AB$, intersecting the semicircle at $F$ and $G$, respectively. Find the measurement of $\angle{FDG}$.
Let point $O$ be the centroid of $\triangle{ABC}$, prove the areas of $\triangle{ABO}, \triangle{BOC}$, and $\triangle{COA}$ are equal.
Consider triangle ABC and its circumcircle S. Reflect the circle with respect to AB, AC, BC to get three new circles SAB, SBC, and SBC. Show that these three circles intersect at a common point and identify this point.
Given triangle $ABC$, construct equilateral triangle $ABC_1$, $BCA_1$, $CAB_1$ on the outside of $ABC$. Let $P, Q$ denote the midpoints of $C_1A_1$ and $C_1B_1$ respectively. Let $R$ be the midpoint of $AB$.
Prove that triangle $PQR$ is isosceles.
(Napolean's Triangle) Given triangle $ABC$, construct an equilateral triangle on the outside of each of the sides. Let $P, Q, R$ be the centroids of these equilateral triangles, prove that triangle $PQR$ is equilateral.
Let $W_1W_2W_3$ be a triangle with circumcircle $S$, and let $A_1, A_2, A_3$ be the midpoints of $W_2W_3, W_1W_3,
W_1W_2$ respectively. From Ai drop a perpendicular to the line tangent to $S$ at $W_i$. Prove that these perpendicular lines are concurrent and identify this point of concurrency.
Let $A_0A_1A_2A_3A_4A_5A_6$ be a regular 7-gon. Prove that $$\frac{1}{A_0A_1} = \frac{1}{A_0A_2}+\frac{1}{A_0A_3}$$
Given point $P_0$ in the plane of triangle $A_1A_2A_3$. Denote $A_s = A_{s-3}$, for $s > 3$. Construct points $P_1; P_2; \cdots$ sequentially such that point $P_{k+1}$ is $P_k$ rotated $120^\circ$ counter-clockwise around $A_{k+1}$. Prove that if $P_{1986} = P_0$ then triangle $A_1A_2A_3$ is isosceles.
Point $H$ is the orthocenter of triangle $ABC$. Points $D, E$ and $F$ lie on the circumcircle of triangle $ABC$ such that $AD\parallel BE\parallel CF$. Points $S, T,$ and $U$ are the respective reflections of $D, E, F$ across the
lines $BC, CA$ and $AB$. Prove that $S, T, U, H$ are cyclic.
Let $ABCD$ be a cyclic quadrilateral. Let $P, Q, R$ be the feet of the perpendiculars from $D$ to the lines $BC, CA$ and $AB$ respectively. Show that $PQ = QR$ iff the bisectors of $\angle ABC$ and $\angle ADC$ meet on $AC$.
Let $O$ be the circumcentre of triangle $ABC$. A line through $O$ intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Let $S$ and $R$ be the midpoints of $BN$ and $CM$, respectively. Prove that $\angle ROS = \angle BAC$.