Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle APB> 90^o$ and $\angle PAD + \angle PBD = \angle PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle TRB = \angle DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle PAR$. Prove that $RS = RT$.

Let $P_1P_2 \ldots P_8$ be a convex octagon. An integer $i$ is chosen uniformly at random from $1$ to $7$, inclusive. For each vertex of the octagon, the line between that vertex and the vertex $i$ vertices to the right is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?

Show that there are infinitely many composite numbers $n$ such that $3^{n-1}-2^{n-1}$ is divisible by $n$.

A group of 8 players played several tennis tournaments between themselves using the single-elimination system, that is, the players are randomly split into pairs, the winners split into two pairs that play in semifinals, the winners of semifinals play in the final round. It so happened that after several tournaments each player had played with each other exactly once. Prove that [list=a] [*]each player participated in semifinals more than once; [*]each player participated in at least one final. [/list] [i]Boris Frenkin[/i]

Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$

Given $n \geq 3$ pairwise different prime numbers $p_1, p_2, ....,p_n$. Given, that for any $k \in \{ 1,2,....,n \}$ residue by division of $ \prod_{i \neq k} p_i$ by $p_k$ equals one number $r$. Prove, that $r \leq n-2 $.

Show how to dissect a square into at most five pieces in such a way that the pieces can be reassembled to form three squares of (pairwise) distinct areas.

Let $(a_n )_{n\ge 1}$ be a sequence of integers that satisfies \[a_n = a_{n-1} -\text{min}(a_{n-2} , a_{n-3} )\] for all $n \ge 4$. Prove that for every positive integer $k$, there is an $n$ such that $a_n$ is divisible by $3^k$ .

There are $7$ balls in a jar, numbered from $1$ to $7$, inclusive. First, Richard takes $a$ balls from the jar at once, where $a$ is an integer between $1$ and $6$, inclusive. Next, Janelle takes $b$ of the remaining balls from the jar at once, where $b$ is an integer between $1$ and the number of balls left, inclusive. Finally, Tai takes all of the remaining balls from the jar at once, if any are left. Find the remainder when the number of possible ways for this to occur is divided by $1000$, if it matters who gets which ball. [i]Proposed by firebolt360 & DeToasty3[/i]

For polynomial $P(x)=1-\frac{1}{3}x+\frac{1}{6}x^2$, define \[ Q(x) = P(x)P(x^3)P(x^5)P(x^7)P(x^9) = \sum\limits_{i=0}^{50}a_ix^i. \] Then $\sum\limits_{i=0}^{50}|a_i|=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

If $x \neq 0$, $\frac x{2} = y^2$ and $\frac{x}{4} = 4y$, then $x$ equals $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 128 $

Around the acute-angled triangle $ABC$ ($AC>CB$) a circle is circumscribed, and the point $N$ is midpoint of the arc $ACB$ of this circle. Let the points $A_1$ and $B_1$ be the feet of perpendiculars on the straight line $NC$, drawn from points $A$ and $B$ respectively (segment $NC$ lies inside the segment $A_1B_1$). Altitude $A_1A_2$ of triangle $A_1AC$ and altitude $B_1B_2$ of triangle $B_1BC$ intersect at a point $K$ . Prove that $\angle A_1KN=\angle B_1KM$, where $M$ is midpoint of the segment $A_2B_2$ .

During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^3,z^5,z^7,\ldots,z^{2013}$ in that order; on Sunday, he begins at $1$ and delivers milk to houses located at $z^2,z^4,z^6,\ldots,z^{2012}$ in that order. Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^2$.

Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.

Let $m \leq n$ be natural numbers. Starting with the product $t=m\cdot (m+1) \cdot (m+2) \cdot \cdots \cdot n$, let $T_{m, n}$ be the sum of products that can be obtained from deleting from $t$ pairs of consecutive integers (this includes $t$ itself). In the case where all the numbers are deleted, we assume the number $1$. For example, $T_{2, 7} = 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 + 2 \cdot 3 \cdot 4 \cdot 5 + 2 \cdot 3 \cdot 4 \cdot 7 + 2 \cdot 3 \cdot 6 \cdot 7 + 2 \cdot 5 \cdot 6 \cdot 7 + 4 \cdot 5 \cdot 6 \cdot 7 + 2 \cdot 3 + 2 \cdot 5 + 2 \cdot 7 + 4 \cdot 7 + 6 \cdot 7 + 1 = 5040 + 120 + 168 + 252 + 420 + 840 + 6 + 10 + 14 + 20 + 28 + 42 + 1 = 6961$. Taking $T_{n+1, n} = 1$. Show that $T_{m, n+1}=T_{m, k-1} \cdot T_{k+2, n+1} + T_{m, k} \cdot T_{k+1, n+1}$ for all $1 \leq m \leq k \leq n$.

A polynomial $P(x)$ is called [i]nice[/i] if $P(0) = 1$ and the nonzero coefficients of $P(x)$ alternate between $1$ and $-1$ when written in order. Suppose that $P(x)$ is nice, and let $m$ and $n$ be two relatively prime positive integers. Show that \[Q(x) = P(x^n) \cdot \frac{(x^{mn} - 1)(x-1)}{(x^m-1)(x^n-1)}\] is nice as well.

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

Prove that no pair of different positive integers $(m, n)$ exist, such that $\frac{4m^{2}n^{2}-1}{(m^{2}-n^2)^{2}}$ is an integer.

On an infinite (in both directions) strip of squares, indexed by the integers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types: (a) Remove one stone from each of the squares $n - 1$ and $n$ and place one stone on square $n + 1$. (b) Remove two stones from square $n$ and place one stone on each of the squares $n + 1$, $n - 2$. Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves. [i]D. Fon-der-Flaas[/i]

Let $x$ and $y$ be real numbers satisfying the conditions $x + y = 4$ and $xy = 3$. Compute the value of $(x - y)^2$. A. $0$ B. $1$ C. $4$ D. $9$ E.$ -1$

Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.

Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$ be real numbers satisfying the conditions: \[ \left\{\begin{array}{cccc} |x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n | & \equal{} & 1 & \ \\ |x_i| & \leq & \displaystyle \frac {n \plus{} 1}{2} & \ \textrm{ for }i \equal{} 1, 2, \ldots , n. \end{array} \right. \] Show that there exists a permutation $ y_1$, $ y_2$, $ \ldots$, $ y_n$ of $ x_1$, $ x_2$, $ \ldots$, $ x_n$ such that \[ | y_1 \plus{} 2 y_2 \plus{} \cdots \plus{} n y_n | \leq \frac {n \plus{} 1}{2}. \]

Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that: (i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$ (ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$ Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$

Find a function $f$ such that $f(t^2 +t +1) = t$ for all real $t \ge 0$