$x\equiv\left(\sum_{k=1}^{2007}k\right)\mod{2016}$, where $0\le x\le 2015$. Solve for $x$.

Prove that, for any integers $a,b,c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.

Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations \[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]

Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.

We have a chessboard and we call a $1\times1$ square a room. A robot is standing on one arbitrary vertex of the rooms. The robot starts to move and in every one movement, he moves one side of a room. This robot has $2$ memories $A,B$. At first, the values of $A,B$ are $0$. In each movement, if he goes up, $1$ unit is added to $A$, and if he goes down, $1$ unit is waned from $A$, and if he goes right, the value of $A$ is added to $B$, and if he goes left, the value of $A$ is waned from $B$. Suppose that the robot has traversed a traverse (!) which hasn’t intersected itself and finally, he has come back to its initial vertex. If $v(B)$ is the value of $B$ in the last of the traverse, prove that in this traverse, the interior surface of the shape that the robot has moved on its circumference is equal to $|v(B)|$.

Let $n > 1$ be a natural number. There are $n$ bulbs in a line, each of which can be on or off. Every minute, simultaneously, all the lit bulbs turn off and the unlit bulbs that were adjacent to exactly one lit bulb turn on. Determine for what values of $n$ there is an initial arrangement such that if this process is followed indefinitely, all the lights will never be off.

Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$.

Use calculus to find the behaviour of the function \[y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty\] and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima and points of inflection in your graph.

Arthur stands on a circle drawn with chalk in a parking lot. It is sunrise and there are birds in the trees nearby. He stands on one of five triangular nodes that are spaced equally around the circle, wondering if and when the aliens will pick him up and carry him from the node he is standing on. He flips a fair coin $12$ times, each time chanting the name of a nearby star system. Each time he flips a head, he walks around the circle, in the direction he is facing, until he reaches the next node in that direction. Each time he flips a tail, he reverses direction, then walks around the circle until he reaches the next node in that new direction. After $12$ flips, Arthur finds himself on the node at which he started. He thinks this is fate, but Arthur is quite mistaken. If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that Arthur flipped exactly $6$ heads, find $a+b$.

Let $\zeta_1, \zeta_2,\ldots$ be identically distributed, independent real-valued random variables with expected value $0$. Suppose that the $\Lambda (\lambda) := \log \mathbb E \exp (\lambda \zeta_i)$ logarithmic moment-generating function always exists for $\lambda\in\mathbb R$ ($\mathbb E$ is the expected value). Furthermore, let $G\colon\mathbb R \rightarrow \mathbb R$ be a function such that $G(x)\leq \min (|x|, x^2)$. Prove that for small $\gamma >0$ the following sequence is bounded: $$\left\{ \mathbb E \exp \left( \gamma l G \left( \frac 1l (\zeta_1+\ldots + \zeta_l)\right)\right)\right\}^{\infty}_{l=1}$$ (translated by j___d)

Starting with a positive integer $M$ written on the board , Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x+2$.Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he replaces it with $2x+27$.Given that Alice and Bob reach the same number after playing $4$ moves each, find the smallest value of $M+N$

Positive numbers $ x$, $ y$, and $ z$ satisfy $ xyz \equal{} 10^{81}$ and $ (\log_{10}x)(\log_{10} yz) \plus{} (\log_{10}y) (\log_{10}z) \equal{} 468$. Find $ \sqrt {(\log_{10}x)^2 \plus{} (\log_{10}y)^2 \plus{} (\log_{10}z)^2}$.

Let $a,b$ be real numbers satisfying $a^{2} + b^{2} = 3ab = 75$ and $a>b$. Compute $a^{3}-b^{3}$.

Find all functions $f : R \to R$ that satisfy $f((x-y)^2)= f(x)^2 -2x f(y)+y^2$ for all $x,y \in R$

At a party with $n$ people, it is known that among any $4$ people, there are either $3$ people who all know one another or $3$ people none of which knows another. Show that the $n$ people can be separated into two rooms, so that everyone in one room knows one another and no two people in the other room know each other.

Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.

All vertices of a convex polyhedron with 100 edges are cut off by some planes. The planes do not intersect either inside or on the surface of the polyhedron. For this new polyhedron find a) the number of vertices; [i](1 point)[/i] b) the number of edges. [i](2 points)[/i]

Let $ n$ be an integer, $ n \geq 3.$ Let $ a_1, a_2, \ldots, a_n$ be real numbers such that $ 2 \leq a_i \leq 3$ for $ i \equal{} 1, 2, \ldots, n.$ If $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n,$ prove that \[ \frac{a^2_1 \plus{} a^2_2 \minus{} a^2_3}{a_1 \plus{} a_2 \minus{} a_3} \plus{} \frac{a^2_2 \plus{} a^2_3 \minus{} a^2_4}{a_2 \plus{} a_3 \minus{} a_4} \plus{} \ldots \plus{} \frac{a^2_n \plus{} a^2_1 \minus{} a^2_2}{a_n \plus{} a_1 \minus{} a_2} \leq 2s \minus{} 2n.\]

A representation of $\frac{17}{20}$ as a sum of reciprocals $$ \frac{17}{20} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} $$ is called a [i]calm representation[/i] with $k$ terms if the $a_i$ are distinct positive integers and at most one of them is not a power of two. (a) Find the smallest value of $k$ for which $\frac{17}{20}$ has a calm representation with $k$ terms. (b) Prove that there are infinitely many calm representations of $\frac{17}{20}$.

At a mathematical olympiad, eight problems were given to 30 contestants. In order to take the difficulty of each problem into account, the jury decided to assign weights to the problems as follows: a problem is worth $n$ points if it was not solved by exactly $n$ contestants. For example, if a problem was solved by all contestants, then it is worth no points. (It is assumed that there are no partial marks for a problem.) Ivan got less points than any other contestant. Find the greatest score he can have.

If $0 < a \leq b$ then$$\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{2(b^2 - a^2)}{(a^2+2)(b^2+2)}\right)\leq \int \limits_a^b \frac{(x^2+1)(x^2+x+1)}{(x^3 + x^2 + 1) (x^3 + x + 1)}dx\leq \frac{4}{\sqrt{3}}\tan^{-1}\left(\frac{(b-a)\sqrt{3}}{a+b+2(1+ab)}\right)$$

In a triangle $ABC$, let $A_0$ be the point where the interior angle bisector of angle $A$ meets with side $BC$. Similarly define $B_0$ and $C_0$. Prove that $\angle B_0A_0C_0 = 90^o$ if and only if $\angle BAC = 120^o$.

Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers

Given a circle $\omega$, a point $A$ lying inside $\omega$, and point $B$ ($B \ne A$). All possible triangles $BXY$ are considered, such that the points $X$ and $Y$ lie on $\omega$ and the chord $XY$ passes through the point $A$. Prove that the centers of the circumcircles of the triangles $BXY$ lie on the same straight line.

Let $n{}$ be a natural number. The playing field for a "Master Sudoku" is composed of the $n(n+1)/2$ cells located on or below the main diagonal of an $n\times n$ square. A teacher secretly selects $n{}$ cells of the playing field and tells his student [list] [*]the number of selected cells on each row, and [*]that there is one selected cell on each column. [/list]The teacher's selected cells form a Master Sudoku if his student can determine them with the given information. How many Master Sudokus are there? [i]Proposed by T. Amdeberkhan, M. Ruby and F. Petrov[/i]