Let $ABCDEFGH$ be a regular octagon with side length $\sqrt{60}$. Let $\mathcal{K}$ denote the locus of all points $K$ such that the circumcircles (possibly degenerate) of triangles $HAK$ and $DCK$ are tangent. Find the area of the region that $\mathcal{K}$ encloses.

Let $k>1$. Fix a circle $\omega$ with center $O$ and radius $r$, and fix a point $A$ with $OA=kr$. Let $AB$, $AC$ be tangents to $\omega$. Choose a variable point $P$ on the minor arc $BC$ in $\omega$. Lines $AB$ and $CP$ intersect at $X$ and lines $AC$ and $BP$ intersect at $Y$. The circles $(BPX)$ and $(CPY)$ meet at another point $Z$. Prove that the line $PZ$ always passes through a fixed point except for one value of $k>1$, and determine this value. [i]Proposed by Ivan Chan Kai Chin[/i]

An Annville Junior High School, $30\%$ of the students in the Math Club are in the Science Club, and $80\%$ of the students in the Science Club are in the Math Club. There are $15$ students in the Science Club. How many students are in the Math Club? $ \text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 30\qquad\text{(D)}\ 36\qquad\text{(E)}\ 40 $

Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$

Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.

In the following grid below, each row and column contains the numbers $1,2,3,4,5$ exactly once. Furthermore, each of the three sections have the same sum. Find, with proof, all possible ways to fill the grid in. [asy] unitsize(0.5 cm); draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0),linewidth(3)); draw((1,0)--(1,1)--(0,1)); draw((2,0)--(2,3)--(0,3)); draw((3,0)--(3,2)--(0,2)); draw((2,5)--(2,4)--(5,4)); draw((3,5)--(3,3)--(5,3)); draw((4,5)--(4,2)--(5,2)); draw((4,0)--(4,1)); draw((1,5)--(1,4)); draw((0,4)--(1,4)--(1,1)--(5,1),linewidth(3)); draw((1,4)--(2,4)--(2,3)--(3,3)--(3,2)--(4,2)--(4,1),linewidth(3)); [/asy]

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the point inside triangle $ABC$ with the property that $\overline{BD} \perp \overline{CD}$ and $\overline{AD} \perp \overline{BC}$. Then the length $AD$ can be expressed in the form $m-\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$. [i]Proposed by Michael Ren[/i]

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Consider an urn containing $51$ white and $50$ black balls. Every turn, we randomly pick a ball, record the color of the ball, and then we put the ball back into the urn. We stop picking when we have recorded $n$ black balls, where $n$ is an integer randomly chosen from $\{1, 2,... , 100\}$. What is the expected number of turns?

Let $ABC$ be a triangle with $\angle BAC=40^\circ $, $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$. If $AD\parallel OG$, find $\angle ACB$.

It is known that $\cos 157^o = a$, where $a$ is given. Calculate $1^o$ in terms of $a$.

For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$

For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge\frac{8y\sqrt{xy}}{3\sqrt{3}}.$

Kathryn has a crush on Joe. Dressed as Catwoman, she attends the same school Halloween party as Joe, hoping he will be there. If Joe gets beat up, Kathryn will be able to help Joe, and will be able to tell him how much she likes him. Otherwise, Kathryn will need to get her hipster friend, Max, who is DJing the event, to play Joe’s favorite song, “Pieces of Me” by Ashlee Simpson, to get him out on the dance floor, where she’ll also be able to tell him how much she likes him. Since playing the song would be in flagrant violation of Max’s musical integrity as a DJ, Kathryn will have to bribe him to play the song. For every $\$10$ she gives Max, the probability of him playing the song goes up $10\%$ (from $0\%$ to $10\%$ for the first $\$10$, from $10\%$ to $20\%$ for the next $\$10$, all the way up to $100\%$ if she gives him $\$100$). Max only accepts money in increments of $\$10$. How much money should Kathryn give to Max to give herself at least a $65\%$ chance of securing enough time to tell Joe how much she likes him?

A positive integer $a > 1$ is given (in decimal notation). We copy it twice and obtain a number $b = \overline{aa}$ which happens to be a multiple of $a^2$. Find all possible values of $b/a^2$.

$H$ is the orthocentre of an acute–angled triangle $ABC$. A point $E$ is taken on the line segment $CH$ such that $ABE$ is a right–angled triangle. Prove that the area of the triangle $ABE$ is the geometric mean of the areas of triangles $ABC$ and $ABH$.

Determine all real solutions of the equation: \[{ \frac{x^{2}}{x-1}+\sqrt{x-1}+\frac{\sqrt{x-1}}{x^{2}}}=\frac{x-1}{x^{2}}+\frac{1}{\sqrt{x-1}}+\frac{x^{2}}{\sqrt{x-1}} . \]

It is known that $x^2_1+ x^2_2+...+ x^2_6= 6$ and $x_1 + x_2 +....+ x_6 = 0.$ Prove that $ x_1x_2....x_6 \le \frac12$ . .

Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

Find all real numbers $c$ for which there exists a nonconstant two-variable polynomial $P(x, y)$ with real coefficients satisfying \[[P(x, y)]^2 = P(cxy, x^2 + y^2)\] for all real $x$ and $y$. [i]Nikolai Beluhov and Konstantin Garov[/i]

Prove that the sum of the distances from any point in space from the vertices of a cube with edge $a$ is not less than $4\sqrt3 a$.

Let $(\Omega, \mathcal{A},\mathbb{P})$ be a standard probability space, and $\mathcal{X}$ be the set of all bounded random variables. For $t>0$, defined the mapping $R_t$ by \[R_t(X)=t\log \mathbb{E}[\exp(X/t)], \quad X \in \mathcal{X},\] and for $t \in (0,1)$ define the mapping $Q_t$ by \[Q_t(X)=\inf\{x \in \mathbb{R}: \mathbb{P}(X>x) \le t\}, \quad X \in \mathcal{X}.\] For two mappings $f,g:\mathcal{X} \to [-\infty,\infty)$, defined the operator $\square$ by \[f\square g(X)=\inf\{f(Y)+g(X-Y): Y \in \mathcal{X}\}, \quad X \in \mathcal{X}.\] (a) Show that, for $t,s>0$, \[R_t \square R_s=R_{t+s}.\] (b) Show that, for $t,s>0$ with $t+s<1$, \[Q_t \square Q_s=Q_{t+s}.\]

Let $ A,B,C,D,E$ be five distinct points, such that no three of them lie on the same line. Prove that \[ AB\plus{}BC\plus{}CA \plus{} DE < AD \plus{} AE \plus{} BD\plus{}BE \plus{} CD\plus{}CE .\]

A circle is tangent to both branches of the hyperbola $x^2 - 20y^2 = 24$ as well as the $x$-axis. Compute the area of this circle.

How many solutions does \[ x^2 - [x^2] = \left(x - [x]\right)^2 \] have satisfying $1 \leq x \leq n$?