Find all $x \in R$ such that $$ x - \left[ \frac{x}{2016} \right]= 2016$$, where $[k]$ represents the largest smallest integer or equal to $k$.

Let $(a_n )_{n \ge o}$ and $(b_n )_{n \ge o}$ be sequences defined by $a_{n+2} = a_{n+1}+ a_n$ , $n = 0 , 1 , . .. $, $a_0 = 1$, $a_1 = 2$, and $b_{n+2} = b_{n+1} + b_n$ , $n = 0 , 1 , . . .$, $b_0 = 2$, $b_1 = 1$. How many integers do the sequences have in common?

Determine all positive integers $ n$, for which $ 2^{n\minus{}1}n\plus{}1$ is a perfect square.

The eighth grade class at Lincoln Middle School has $93$ students. Each student takes a math class or a foreign language class or both. There are $70$ eighth graders taking a math class, and there are $54$ eighth graders taking a foreign language class. How many eighth graders take [i]only[/i] a math class and [i]not[/i] a foreign language class? $\textbf{(A) }16\qquad \textbf{(B) }23\qquad \textbf{(C) }31\qquad \textbf{(D) }39\qquad \textbf{(E) }70\qquad$

Suppose that the $n$ times differentiable real function $f(x)$ has at least $n+1$ distinct zeros in the closed interval $[a,b]$ and that the polynomial $P(z)=z^n +c_{n-1}z^{n-1}+\ldots+c_1 x +c_0$ has only real zeroes. Show that $f^{(n)}(x)+ c_{n-1} f^{(n-1)}(x) +\ldots +c_1 f'(x)+ c_0 f(x)$ has at least one zero in $[a,b]$, where $f^{(n)}$ denotes the $n$-th derivative of $f.$

Prove that $$AB + PQ + QR + RP \le AP + AQ + AR + BP + BQ + BR$$ where $A, B, P, Q$ and $R $ are any five points in a plane.

Solve the equation, $$\sqrt{x+5}+\sqrt{16-x^2}=x^2-25$$

Collinear points $A$, $B$, and $C$ are given in the Cartesian plane such that $A= (a, 0)$ lies along the x-axis, $B$ lies along the line $y=x$, $C$ lies along the line $y=2x$, and $\frac{AB}{BC}=2$. If $D= (a, a)$, and the circumcircle of triangle $ADC$ intersects the line $y=x$ again at $E$, and ray $AE$ intersects $y=2x$ at $F$, evaluate $\frac{AE}{EF}$.

Let $n\in\mathbb{N}, x_0=0$ and $x_1,x_2,\ldots,x_n$ be postive real numbers such that $x_1+x_2+\ldots+x_n=1$. Show that $$1\leq\sum_{i=1}^{n}\frac{x_i}{\sqrt{1+x_0+x_1+\ldots+x_{i-1}}\cdot\sqrt{x_i+x_{i+1}+\ldots+x_n}}<\frac{\pi}{2}.$$

We are given a natural number $k$. Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute $n$ coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves: $(i)$ We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other. $(ii)$ We choose a field with at least $2$ coins on it, and we transfer one coin from the chosen field to the $k-\mathrm{th}$ field on the left , and one coin from the chosen field to the $k-\mathrm{th}$ field on the right. $\mathbf{(a)}$ If $n\leq k+1$, prove that we can play only finitely many moves. $\mathbf{(b)}$ For which values of $k$ we can choose a natural number $n$ and distribute $n$ coins on the given board such that we can play infinitely many moves.

An array $ (a_1,a_2,...,a_{13})$ of $ 13$ integers is called $ tame$ if for each $ 1 \le i \le 13$ the following condition holds: If $ a_i$ is left out, the remaining twelve integers can be divided into two groups with the same sum of elements. A tame array is called $ turbo$ $ tame$ if the remaining twelve numbers can always be divided in two groups of six numbers having the same sum. $ (a)$ Give an example of a tame array of $ 13$ integers (not all equal). $ (b)$ Prove that in a tame array all numbers are of the same parity. $ (c)$ Prove that in a turbo tame array all numbers are equal.

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?

Let the circles $C_1, C_2$, and $C_3$ be orthogonal to the circle $C$ and intersect each other inside $C$ forming acute angles of measures $A, B$, and $C$. Show that $A + B +C < \pi.$

How many $5\times5$ grids are possible such that each element is either $0$ or $1$ and each row sum and column sum is $4$? $\textbf{(A)}~64$ $\textbf{(B)}~32$ $\textbf{(C)}~120$ $\textbf{(D)}~96$

$a,b$ are positive integers and $(a!+b!)|a!b!$.Prove that $3a\ge 2b+2$.

Irja and Valtteri are tossing coins. They take turns, Irja starting. Each of them has a pebble which reside on opposite vertices of a square at the start. If a player gets heads, she or he moves her or his pebble on opposite vertex. Otherwise the player in turn moves her or his pebble to an adjacent vertex so that Irja proceeds in positive and Valtteri in negative direction. The winner is the one who can move his pebble to the vertex where opponent's pebble lies. What is the probability that Irja wins the game?

Let $A,B,C,D$ be four arbitrary distinct points in space. $(a)$ Prove that using the segments $AB +CD, AC +BD$ and $AD +BC$, it is always possible to construct a triangle $T$ that is non-degenerate and has no obtuse angle. $(b)$ What should these four points satisfy in order for the triangle $T$ to be right-angled?

There are $16$ points marked on the circle. Find the greatest possible number of acute triangles with vertices at the marked points.

Say a positive integer $n>1$ is $d$-coverable if for each non-empty subset $S\subseteq \{0, 1, \ldots, n-1\}$, there exists a polynomial $P$ with integer coefficients and degree at most $d$ such that $S$ is exactly the set of residues modulo $n$ that $P$ attains as it ranges over the integers. For each $n$, find the smallest $d$ such that $n$ is $d$-coverable, or prove no such $d$ exists. [i]Proposed by Carl Schildkraut[/i]

How many ways can a set of $ n $ items be partitioned into two sets?

The angle bisector $[AD]$, the median $[BM]$ and the height $[CH]$ of the acute-angled triangle $ABC$ intersect in one point. Prove that the $\angle BAC> 45^o$.

For each positive integer $n$, let $f_n$ be a real-valued symmetric function of $n$ real variables. Suppose that for all $n$ and all real numbers $x_1,\ldots,x_n, x_{n+1},y$ it is true that $\;(1)\; f_{n}(x_1 +y ,\ldots, x_n +y) = f_{n}(x_1 ,\ldots, x_n) +y,$ $\;(2)\;f_{n}(-x_1 ,\ldots, -x_n) =-f_{n}(x_1 ,\ldots, x_n),$ $\;(3)\; f_{n+1}(f_{n}(x_1,\ldots, x_n),\ldots, f_{n}(x_1,\ldots, x_n), x_{n+1}) =f_{n+1}(x_1 ,\ldots, x_{n}).$ Prove that $f_{n}(x_{1},\ldots, x_n) =\frac{x_{1}+\cdots +x_{n}}{n}.$

Let $M$ be the intersection of the medians $ABC$ of the triangle and the midpoint of the side $BC$. $A$ line parallel to side $BC$ and passing through point $M$ intersects sides $AB$ and $AC$ at points $X$ and $Y$ respectively. Let the point of intersection of the lines $XC$ and $MB$ be $Q$ and let $P$ intersection point of the lines $YB$ and $MC$ be $P$ . Prove that the triangles $DPQ$ and $ABC$ are similar.

In a group of 12 persons, among any 9 there are 5 which know each other. Prove that there are 6 persons in this group which know each other

Let $f:[0,1]\to\mathbb{R}$ be a continuous function. Prove that \[\lim_{n\to\infty}\int_0^1 f(x^n) \ dx=f(0).\]Furthermore, if $f(0)=0$ and $f$ is right-differentiable in $0{}$, prove that the limits \[\lim_{\varepsilon\to0}\int_\varepsilon^1\frac{f(x)}{x} \ dx\quad\text{and}\quad\lim_{n\to\infty}\left(n\int_0^1f(x^n) \ dx\right)\]exist, are finite and are equal.