In a triangle $ ABC$, the bisector of $ \angle ACB$ cuts the side $ AB$ at $ D$. An arbitrary circle $ (O)$ passing through $ C$ and $ D$ meets the lines $ BC$ and $ AC$ at $ M$ and $ N$ (different from $ C$), respectively. (a) Prove that there is a circle $ (S)$ touching $ DM$ at $ M$ and $ DN$ at $ N$. (b) If circle $ (S)$ intersects the lines $ BC$ and $ CA$ again at $ P$ and $ Q$ respectively, prove that the lengths of the segments $ MP$ and $ NQ$ are constant as $ (O)$ varies.

On a real number line, the points $1, 2, 3, \dots, 11$ are marked. A grasshopper starts at point $1$, then jumps to each of the other $10$ marked points in some order so that no point is visited twice, before returning to point $1$. The maximal length that he could have jumped in total is $L$, and there are $N$ possible ways to achieve this maximum. Compute $L+N$. [i]Proposed by Yannick Yao[/i]

It is known that there are $3$ buildings in the same shape which are located in an equilateral triangle. Each building has a $2015$ floor with each floor having one window. In all three buildings, every $1$st floor is uninhabited, while each floor of others have exactly one occupant. All windows will be colored with one of red, green or blue. The residents of each floor of a building can see the color of the window in the other buildings of the the same floor and one floor just below it, but they cannot see the colors of the other windows of the two buildings. Besides that, sresidents cannot see the color of the window from any floor in the building itself. For example, resident of the $10$th floor can see the colors of the $9$th and $10$th floor windows for the other buildings (a total of $4$ windows) and he can't see the color of the other window. We want to color the windows so that each resident can see at lest $1$ window of each color. How many ways are there to color those windows?

Let $m,n,k$ be pairwise relatively prime positive integers greater than $3$. Find the minimal possible number of points on the plane with the following property: there are $x$ of them which are the vertices of a regular $x$-gon for $x = m, x = n, x = k$. (E.Piryutko)

A $20 \times 20$ treasure map is glued to a torus. A treasure is hidden in a cell of this map. We can ask questions about $1\times 4$ or $4 \times 1$ rectangles so that we find out if there is a treasure in this rectangle or not. The answers to all questions are absolutely true, but they are given only after all rectangles we want to ask are set. What is the least amount of questions needed to be asked so that we can be sure to find the treasure? (If you describe the position of the cells in a torus with numbers $(i, j)$ of row and column, $1 \leq i, j \leq 20$, then two cells are neighbors, if and only if two of the coordinates they have are the same, and the other two differ by $1$ mod $20$.)

Let $n\geqslant 2$ and $a_1,a_2,\ldots,a_n$ be non-zero integers such that $a_1+a_2+\cdots+a_n=a_1a_2\cdots a_n.$ Prove that \[(a_1^2-1)(a_2^2-1)\cdots(a_n^2-1)\]is a perfect square.

The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.

Two kids $A$ and $B$ play a game as follows: from a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that: 1. $A$ goes first. 2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n - 1$, inclusive. 3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive. The winner is the one who takes the last marble. Determine all natural numbers $n$ for which $A$ has a winning strategy

Show that the only solutions of te equation \[ p^{k} + 1 = q^{m} \], in positive integers $k,q,m > 1$ and prime $p$ are (i) $(p,k,q,m) = (2,3,3,2)$ (ii) $k=1 , q=2,$and $p$ is a prime of the form $2^{m} -1$, $m > 1 \in \mathbb{N}$

Let \( I \) and \( O \) be the incenter and circumcenter of the right triangle \( ABC \) (\( \angle C = 90^\circ \)), and let \( K \) be the tangency point of the incircle with \( AC \). Let \( P \) and \( Q \) be the points where the circumcircle of triangle \( AOK \) intersects \( OC \) and the circumcircle of triangle \( ABC \), respectively. Prove that points \( C, I, P, \) and \( Q \) are concyclic. [i]Proposed by Mykhailo Sydorenko[/i]

Let $M, N, P$ be the midpoints of the sides $AB, BC, CA$ of a triangle $ABC$. Let $X$ be a fixed point inside the triangle $MNP$. The lines $L_1, L_2, L_3$ that pass through point $X$ are such that $L_1$ intersects segment $AB$ at point $C_1$ and segment $AC$ at point $B_2$; $L_2$ intersects segment $BC$ at point $A_1$ and segment $BA$ at point $C_2$; $L_3$ intersects segment $CA$ at point $B_1$ and segment $CB$ at point $A_2$. Indicates how to construct the lines $L_1, L_2, L_3$ in such a way that the sum of the areas of the triangles $A_1A_2X, B_1B_2X$ and $C_1C_2X$ is a minimum.

Let $M$ be the midpoint of a given segment $BC$. Point $A$ is chosen to maximize $\angle ABC$ while subject to the condition that $\angle MAC = 20^o$ . What is the ratio $BC/BA$ ?

Given the vertex $A$ and the centroid $M$ of a triangle $ABC$, find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ, 70^\circ].$

Let $N_5$ be the answer to problem 5. Triangle $JHU$ satisfies $JH=N_5$ and $JU=6$. Point $X$ lies on $\overline{HU}$ such that $\overline{JX}$ is an altitude of $\triangle{JHU}$, point $Y$ is the midpoint of $\overline{JU}$, and $\overline{JX}$ and $\overline{HY}$ intersect at $Z$. Assume that $\triangle{HZX}$ is similar to $\triangle{JZY}$ (in this vertex order). Compute the area of $\triangle{JHU}$.

A positive number is mistakenly divided by $6$ instead of being multiplied by $6$. Based on the correct answer, the error thus comitted, to the nearest percent, is: $\textbf{(A)}\ 100 \qquad \textbf{(B)}\ 97 \qquad \textbf{(C)}\ 83 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 3 $

Mindy made three purchases for $ \$1.98, \$5.04$ and $ \$9.89$. What was her total, to the nearest dollar? $ \textbf{(A)}\ \$10 \qquad \textbf{(B)}\ \$15 \qquad \textbf{(C)}\ \$16 \qquad \textbf{(D)}\ \$17 \qquad \textbf{(E)}\ \$18$

For some natural numbers $a, b, c$ and $d$ the following equations holds: $$\frac{a}{c}= \frac{b}{d}= \frac{ab + 1}{cd + 1} .$$ Prove that $a = c$ and $b = d$.

Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$. [i]Alex Zhu.[/i]

Let $ABC$ be a triangle, right-angled at point $ A$ and with $AB>AC$. The tangent through $ A$ of the circumcircle $G$ of $ABC$ cuts $BC$ at $D$. $E$ is the reflection of $ A$ over line $BC$. $X$ is the foot of the perpendicular from $ A$ over $BE$. $Y$ is the midpoint of $AX$, $Z$ is the intersection of $BY$ and $G$ other than $ B$, and $F$ is the intersection of $AE$ and $BC$. Prove $D, Z, F, E$ are concyclic.

$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.

You have a $10\times 10$ grid of squares. You write a number in each square as follows: you write $1$, $2$, $3$ ,$ ...$ ,$10$ from left to right across the top row, then $11$, $12$, $...$, $20$ across the second row, and so on, ending with a $100$ in the bottom right square. You then write a second number in each square, writing $1$, $2$, $3$ ,$ ...$ ,$10$ in the first column (from top to bottom), then $11$, $12$, $...$, $20$ in the second column, and so forth. When this process is finished, how many squares will have the property that their two numbers sum to $101$?

Find all reals $\lambda$ for which there is a nonzero polynomial $P$ with real coefficients such that $$\frac{P(1)+P(3)+P(5)+\ldots+P(2n-1)}n=\lambda P(n)$$for all positive integers $n$, and find all such polynomials for $\lambda=2$.

Consider an acute triangle $ABC$ and it's circumcircle $\omega$. With center $A$, we construct a circle $\gamma$ that intersects arc $AB$ of circle $\omega$ , that doesn't contain $C$, at point $D$ and arc $AC$ , that doesn't contain $B$, at point $E$. Suppose that the intersection point $K$ of lines $BE$ and $CD$ lies on circle $\gamma$. Prove that line $AK$ is perpendicular on line $BC$.

A set $S$ consists of $k$ sequences of $0,1,2$ of length $n$. For any two sequences $(a_i),(b_i)\in S$ we can construct a new sequence $(c_i)$ such that $c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor$ and include it in $S$. Assume that after performing finitely many such operations we obtain all the $3n$ sequences of $0,1,2$ of length $n$. Find the least possible value of $k$.

Let $A$ be a real number and $(a_{n})$ be a sequence of real numbers such that $a_{1}=1$ and \[1<\frac{a_{n+1}}{a_{n}}\leq A \mbox{ for all }n\in\mathbb{N}.\] $(a)$ Show that there is a unique non-decreasing surjective function $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $1<A^{k(n)}/a_{n}\leq A$ for all $n\in \mathbb{N}$. $(b)$ If $k$ takes every value at most $m$ times, show that there is a real number $C>1$ such that $Aa_{n}\geq C^{n}$ for all $n\in \mathbb{N}$.