In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$.

The frame of a painting has the form of a $105^{\prime\prime}$ by $105^{\prime\prime}$ square with a $95^{\prime\prime}$ by $95^{\prime\prime}$ square removed from its center. The frame is built out of congruent isosceles trapezoids with angles measuring $45^\circ$ and $135^\circ$. Each trapezoid has one base on the frame's outer edge and one base on the frame's inner edge. Each outer edge of the frame contains an odd number of trapezoid bases that alternate long, short, long, short, etc. What is the maximum possible number of trapezoids in the frame?

In the triangle $ABC$, the orthocenter $H$ lies on the inscribed circle. Is this triangle necessarily isosceles?

Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand? $\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$

Let $ABCD$ be a trapezoid with bases $BC \parallel AD$, where $AD > BC$, and non-parallel legs $AB$ and $CD$. Let $M$ be the intersection of $AC$ and $BD$. Let $\Gamma_1$ be a circumference that passes through $M$ and is tangent to $AD$ at point $A$; let $\Gamma_2$ be a circumference that passes through $M$ and is tangent to $AD$ at point $D$. Let $S$ be the intersection of the lines $AB$ and $CD$, $X$ the intersection of $\Gamma_1$ with the line $AS$, $Y$ the intesection of $\Gamma_2$ with the line $DS$, and $O$ the circumcenter of triangle $ASD$. Show that $SO \perp XY$.

In a non-isosceles triangle $ABC$, let $I$ be its incenter. The incircle of $ABC$ touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A line passing through $D$ and perpendicular to $AD$ intersects $IB$ and $IC$ at $A_b$ and $A_c$, respectively. Define the points $B_c$, $B_a$, $C_a$, and $C_b$ similarly. Let $G$ be the intersection of the cevians $AD$, $BE$, and $CF$. The points $O_1$ and $O_2$ are the circumcenter of the triangles $A_bB_cC_a$ and $A_cB_aC_b$, respectively. Prove that $IG$ is the perpendicular bisector of $O_1O_2$.

Consider a circle $k$ with center $S$ and radius $r$. Let a point $A\neq S$ be given with $SA=d<r$. Consider a light ray emitted at point $A$, reflected at point $B\in k$, further reflected in point $C\in k$, which then passes through the original point $A$. Compute the sinus of convex angle $SAB$ in terms of $d,r$ and discuss conditions of solvability.

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered? $\text{(A)}\ 71 \qquad\text{(B)}\ 76 \qquad\text{(C)}\ 80 \qquad\text{(D)}\ 82 \qquad\text{(E)}\ 91$

Compute the following: $$\sum^{99}_{x=0} (x^2 + 1)^{-1} \,\,\, (mod \,\,\,199)$$ where $x^{-1}$ is the value $0 \le y \le 199$ such that $xy - 1$ is divisible by $199$.

Find all functions $f$ from the reals to the reals such that \[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\] for all real $x,y$.

Consider a cake in the shape of a circle. It's been divided to some inequal parts by its radii. Arash and Bahram want to eat this cake. At the very first, Arash takes one of the parts. In the next steps, they consecutively pick up a piece adjacent to another piece formerly removed. Suppose that the cake has been divided to 5 parts. Prove that Arash can choose his pieces in such a way at least half of the cake is his.

There exists one pair of positive integers $ a, b $ such that $ 100 > a > b > 0 $ and $ \dfrac{1}{a} + \dfrac{1}{b} = \dfrac{2}{35} $. Find $ a + b $.

Let $ABCD$ be a cyclic quadrilateral. Let $X$ be the foot of the perpendicular from the point $A$ to the line $BC$, let $Y$ be the foot of the perpendicular from the point $B$ to the line $AC$, let $Z$ be the foot of the perpendicular from the point $A$ to the line $CD$, let $W$ be the foot of the perpendicular from the point $D$ to the line $AC$. Prove that $XY\parallel ZW$. Darij

A circle with area $\tfrac{36}{\pi}$ has the same perimeter as a square with what side length? $\textbf{(A) }\frac{9}{\pi}\qquad\textbf{(B) }3\qquad\textbf{(C) }\pi\qquad\textbf{(D) }6\qquad\textbf{(E) }\pi^2$

Let $f$ be a function defined for the non-negative integers, such that: a) $f(n)=0$ if $n=2^{j}-1$ for some $j \geq 0$. b) $f(n+1)=f(n)-1$ otherwise. i) Show that for every $n \geq 0$ there exists $k \geq 0$ such that $f(n)+n=2^{k}-1$. ii) Find $f(2^{1990})$.

How many multiples of $2005$ are factors of $(2005)^2$?

Answer the following questions. (1) For $x\geq 0$, show that $x-\frac{x^3}{6}\leq \sin x\leq x.$ (2) For $x\geq 0$, show that $\frac{x^3}{3}-\frac{x^5}{30}\leq \int_0^x t\sin t\ dt\leq \frac{x^3}{3}.$ (3) Find the limit \[\lim_{x\rightarrow 0} \frac{\sin x-x\cos x}{x^3}.\]

Prove that $n^2 + 8n + 15$ is not divisible by $n + 4$ for any positive integer $n$.

Régis, Ed and Rafael are at the IMO. They are going to play a game in Bath, and there are $2^n$ houses in the city. Régis and Ed will team up against Rafael. The game operates as follows: First, Régis and Ed think on a strategy and then let Rafael know it. After this, Régis and Ed no longer communicate, and the game begins. Rafael decides on an order to visit the houses and then starts taking Régis to them in that order. At each house, except for the last one, Régis choose a number between $1$ and $n$ and places it in the house. In the last house, Rafael chooses a number from $1$ to $n$ and places it there. Afterwards, Ed sees all the houses and the numbers in them, and he must guess in which house Rafael placed the number. Ed is allowed $k$ guesses. What is the smallest $k$ for which there exists a strategy for Ed and Régis to ensure that Ed correctly guess the house where Rafael placed the number?

$s,t\in\mathbb{R}$, then the minumum value of $(s+5-3|\cos t|)^2+(s-2|\sin t|)^2$ is________.

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

Let $ ABCD$ be a cyclic quadrilateral and $ r$ and $ s$ the lines obtained reflecting $ AB$ with respect to the internal bisectors of $ \angle CAD$ and $ \angle CBD$, respectively. If $ P$ is the intersection of $ r$ and $ s$ and $ O$ is the center of the circumscribed circle of $ ABCD$, prove that $ OP$ is perpendicular to $ CD$.

Let $O $, $O_1$, $O_2 $, $O_3$, $O_4$ be points such that $O_1$, $O$, $O_3$ and $O_2$, $O$, $O_4$ are collinear in that order, $OO_1 =1$, $OO_2 = 2$, $OO_3 =\sqrt2$, $OO_4 = 2$, and $\angle O_1OO_2 = 45^o$. Let $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ be the circles with respective centers $O_1$, $O_2$ , $O_3$, $O_4$ that go through $O$. Let $A$ be the intersection of $\omega_1$ and $\omega_2$, $B$ be the intersection of $\omega_2$ and $\omega_3$, $C$ be the intersection of $\omega_3$ and $\omega_4$, and $D$ be the intersection of $\omega_4$ and $\omega_1$ with $A$, $B$, $C$, $D$ all distinct from $O$. What is the largest possible area of a convex quadrilateral $P_1P_2P_3P_4$ such that $P_i$ lies on $O_i$ and that $A$, $B$, $C$, $D$ all lie on its perimeter?

I plotted the graphs $y=(x-0)^2, y=(x-5)^2, \ldots, y=(x-45)^2.$ I also draw a line $y=k,$ and notice that it intersects the parabolas at exactly $19$ distinct points. What is $k$?