Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pariwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum_{i=1}^\infty C_{n-2}C_{n-1} = 6p$. Find $p$.
Triangle $ABC$ is inscribed in circle $\omega$. Points $P$ and $Q$ are on side $\overline{AB}$ with $AP < AQ$. Rays $CP$ and $CQ$ meet $\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
(Weitzenbock's Inequality) Let $a, b, c$, and $S$ be a triangle's three sides' lengths and its area, respectively. Show that $$a^2 + b^2 + c^2 \ge 4\sqrt{3}\cdot S$$
Let $ABCD$ be a rectangle where $AB=3$ and $AD=4$. Point $P$ is on the side $AD$. If points $E$ and $F$ are on $AC$ and $BD$ respectively such that $PE \perp AC$ and $PF \perp BD$. Compute $PE+PF$.
We are given a triangle with the following property: one of its angles is quadrisected (divided into four equal angles) by the height, the angle bisector, and the median from that vertex. This property uniquely determines the triangle (up to scaling). Find the measure of the quadrisected angle.
Two sides of a triangle are 4 and 9; the median drawn to the third side has length 6. Find the length of the third side.
A right triangle has legs $a$ and $b$ and hypotenuse $c$. Two segments from the right angle to the hypotenuse are drawn,
dividing it into three equal parts of length $x=\frac{c}{3}$. If the segments have length $p$ and $q$, prove that $p^2 +q^2 =5x^2$.
A circle inscribed in $\triangle{ABC}$ (the incircle) is tangent to $BC$ at $X$, to $AC$ at $Y$ , to $AB$ at $Z$. Show that $AX$, $BY$, and $CZ$ are concurrent.
Three squares are drawn on the sides of $\triangle{ABC}$ (i.e. the square on $AB$ has $AB$ as one of its sides and lies outside $\triangle{ABC}$). Show that the lines drawn from the vertices $A, B, C$ to the centers of the opposite squares are concurrent.
Let $P$ be a point inside a unit square $ABCD$. Find the minimal value of $AP+BP+CP$
The diagonals $AC$ and $CD$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ such that $AM:AC = CN:CE=r$. Determine $r$ if $B, M,$ and $N$ are collinear.
In rectangle $TUVW$, shown here, $WX = 4$ units, $XY = 2$ units, $YV = 1$ unit and $UV = 6$ units. What is the absolute difference between the areas of triangles $TXZ$ and $UYZ$.
Rectangle $ABCD$ is shown with $AB = 6$ units and $AD = 5$ units. If $AC$ is extended to point $E$ such that $AC$ is congruent to $CE$, what is the length of $DE$?
Diagonal $XZ$ of rectangle $WXYZ$ is divided into three segments each of length 2 units by points $M$ and $N$ as shown. Segments $MW$ and $NY$ are parallel and are both perpendicular to $XZ$. What is the area of $WXYZ$? Express your answer in simplest radical form.
On line segment $AE$, shown here, $B$ is the midpoint of segment $AC$ and $D$ is the midpoint of segment $CE$. If $AD = 17$ units and $BE = 21$ units, what is the length of segment $AE$? Express your answer as a common fraction.
A rectangular piece of cardboard measuring 6 inches by 8 inches is trimmed identically on all four corners, as shown, so that each trimmed corner is a quarter circle of greatest possible area. What is the perimeter of the resulting figure? Express your answer in terms of $\pi$.
Spring City is replanting the grass around a circular fountain in the center of the city. The fountain’s diameter is 10 feet, and the grass extends out from the edge of the fountain 20 feet in every direction. Grass seed is sold in bags that will each cover 300 $ft^2$ of grass. How many whole bags of grass seed will the city need to purchase?
Rectangle $ABCD$ is shown with $AB = 6$ units and $AD = 5$ units. If $AC$ is extended to point $E$ such that $AC$ is congruent to $CE$, what is the length of $DE$?
Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.
As shown.
As shown, two vertices of a square are on the circle and one side is tangent to the circle. If the side length of the square is 8, find the radius of the circle.
Triangle $ABC$ is isosceles. An ant begins at $A$, walks exactly halfway along the perimeter of $\triangle{ABC}$, and then returns directly $A$, cutting through the interior of the triangle. The ant's path surround exactly 90% of the area of $\triangle{ABC}$. Compute the maximum value of $\tan{A}$.
In $\triangle{ABC}$, $\angle{BAC} = 40^\circ$ and $\angle{ABC} = 60^\circ$. Points $D$ and $E$ are on sides $AC$ and $AB$, respectively, such that $\angle{DBC}=40^\circ$ and $\angle{ECB}=70^\circ$. Let $F$ be the intersection point of $BD$ and $CE$. Show that $AF\perp BC$.
$DEB$ is a chord of a circle such that $DE=3$ and $EB=5 .$ Let $O$ be the center of the circle. Join $OE$ and extend $OE$ to cut the circle at $C.$ Given $EC=1,$ find the radius of the circle
Chords $AB$ and $CD$ of a given circle are perpendicular to each other and intersect at a right angle at point $E$. If $BE=16$, $DE=4$, and $AD=5$, find $CE$.