Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$, \[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]

Consider the points $A_1$ and $A_2$ on the side $AB$ of the square $ABCD$ taken in such a way that $|AB| = 3 |AA_1| $ and $|AB| = 4 |A_2B|$, similarly consider points $B_1$ and $B_2, C_1$ and $C_2, D_1$ and $D_2$ respectively on the sides $BC$, $CD$ and $DA$. The intersection point of straight lines $D_2A_1$ and $A_2B_1$ is $E$, the intersection point of straight lines $A_2B_1$ and $B_2C_1$ is $F$, the intersection point of straight lines $B_2C_1$ and $C_2D_1$ is $G$ and the intersection point of straight lines $C_2D_1$ and $D_2A_1$ is $H$. Find the area of the square $EFGH$, knowing that the area of $ABCD$ is $1$.

In trapezoid $ ABCD$, $ \overline{AB}$ and $ \overline{CD}$ are perpendicular to $ \overline{AD}$, with $ AB\plus{}CD\equal{}BC$, $ AB<CD$, and $ AD\equal{}7$. What is $ AB\cdot CD$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 12.25 \qquad \textbf{(C)}\ 12.5 \qquad \textbf{(D)}\ 12.75 \qquad \textbf{(E)}\ 13$

Arthur and Renate play a game on a $7 \times 7$ board. Arthur has two red tiles, initially placed on the cells in the bottom left and the upper right corner. Renate has two black tiles, initially placed on the cells in the bottom right and the upper left corner. In a move, a player can choose one of his two tiles and move them to a horizontally or vertically adjacent cell. The players alternate, with Arthur beginning. Arthur wins when both of his tiles are in horizontally or vertically adjacent cells after some number of moves. Can Renate prevent him from winning?

Let $ABC$ be a triangle right in $C$ and the points $D, E$ on the sides $BC$ and $CA$ respectively, such that $\frac{BD}{AC} =\frac{AE}{CD} = k$. Lines $BE$ and $AD$ intersect at $O$. Show that the angle $\angle BOD = 60^o$ if and only if $k =\sqrt3$.

Ed has five identical green marbles and a large supply of identical red marbles. He arranges the green marbles and some of the red marbles in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves equals the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let $m$ be the maximum number of red marbles for which Ed can make such an arrangement, and let $N$ be the number of ways in which Ed can arrange the $m+5$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by 1000.

Two boys $A$ and $B$ start at the same time to ride from Port Jervis to Poughkeepsie, $60$ miles away. $A$ travels $4$ miles an hour slower than $B$. $B$ reaches Poughkeepsie and at once turns back meeting $A$ $12$ miles from Poughkeepsie. The rate of $A$ was: $\textbf{(A)}\ 4\text{ mph}\qquad \textbf{(B)}\ 8\text{ mph} \qquad \textbf{(C)}\ 12\text{ mph} \qquad \textbf{(D)}\ 16\text{ mph} \qquad \textbf{(E)}\ 20\text{ mph}$

Let $k$ be the incircle and let $l$ be the circumcircle of the triangle $ABC$. Prove that for each point $A'$ of the circle $l$, there exists a triangle $(A'B'C')$, inscribed in the circle $l$ and circumscribed about the circle $k.$

Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $ \$4$ per pair and each T-shirt costs $ \$5$ more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $ \$2366$, how many members are in the League? $ \textbf{(A)}\ 77 \qquad \textbf{(B)}\ 91 \qquad \textbf{(C)}\ 143 \qquad \textbf{(D)}\ 182 \qquad \textbf{(E)}\ 286$

There are given integers $a$ and $b$ such that $a$ is different from $0$ and the number $3+ a +b^2$ is divisible by $6a$. Prove that $a$ is negative.

[list=a] [*]The sum of several numbers is equal to one. Can the sum of their cubes be greater than one? [*]The same question as before, for numbers not exceeding one. [*]Can it happen that the series $a_1+a_2+\cdots$ converges, but the series $a_1^3+a_2^3+\cdots$ diverges? [/list]

Find the maximum value of $k$ such that \[\frac{xy}{\sqrt{(x^2 + y^2)(3x^2 + y^2)}}\leq \frac{1}{k}\] holds for all positive numbers $x$ and $y.$

For any prime number $p>2,$ and an integer $a$ and $b,$ if $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{(p-1)^3}=\frac{a}{b},$ prove that $a$ is divisible by $p.$

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Consider a table whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an [i]operation[/i]. It is given that, for infinitely many positive integers $n$, one can obtain, through a finite number of operations, a table having all entries divisible by $n$. Prove that, through a finite number of operations, one can obtain the table whose all entries are zeroes.

Let $ABCD$ be a trapezoid with $AD \parallel BC$, $AD < BC$. Let $E$ be a point on the side $AB$ and $F$ be point on the side $CD$. The circle $(AEF)$ intersects the segment $AD$ again at $A_1$ and the circle $(CEF)$ intersects these segment $BC$ again at $C_1$. Prove that the lines $A_1 C_1$, $BD$ and $EF$ are concurrent. [i]Proposed by A. Kuznetsov[/i]

Determine all triplets $(a,b,c)$ of real numbers such that sets $\{a^2-4c, b^2-2a, c^2-2b \}$ and $\{a-c,b-4c,a+b\}$ are equal and $2a+2b+6=5c$. In every set all elements are pairwise distinct

At a movie theater tickets for adults cost $4$ dollars more than tickets for children. One afternoon the theater sold $100$ more child tickets than adult tickets for a total sales amount of $1475$ dollars. How much money would the theater have taken in if the same tickets were sold, but the costs of the child tickets and adult tickets were reversed?

Let $ABC$ a scalene triangle and $AD, BE, CF$ your angle bisectors, with $D$ in the segment $BC, E$ in the segment $AC$ and $F$ in the segment $AB$. If $\angle AFE = \angle ADC$. Determine $\angle BCA$.

Let $ \left( a_n \right)_{n\ge 1} $ be a sequence of real numbers such that $ a_1>2 $ and $ a_{n+1} =a_1+\frac{2}{a_n} , $ for all natural numbers $ n. $ [b]a)[/b] Show that $ a_{2n-1} +a_{2n} >4 , $ for all natural numbers $ n, $ and $ \lim_{n\to\infty} a_n =2. $ [b]b)[/b] Find the biggest real number $ a $ for which the following inequality is true: $$ \sqrt{x^2+a_1^2} +\sqrt{x^2+a_2^2} +\sqrt{x^2+a_3^2} +\cdots +\sqrt{x^2+a_n^2} > n\sqrt{x^2+a^2}, \quad\forall x\in\mathbb{R} ,\quad\forall n\in\mathbb{N} . $$

Determine all pairs $(a, b)$ of positive integers for which $\frac{a^2b+b}{ab^2+9}$ is an integer number.

One evening a theater sold 300 tickets for a concert. Each ticket sold for \$40, and all tickets were purchased using \$5, \$10, and \$20 bills. At the end of the evening the theater had received twice as many \$10 bills as \$20 bills, and 20 more \$5 bills than \$10 bills. How many bills did the theater receive altogether?

For a positive integer $N$, let $T(N)$ denote the number of arrangements of the integers $1, 2, \cdots N$ into a sequence $a_1, a_2, \cdots a_N$ such that $a_i > a_{2i}$ for all $i$, $1 \le i < 2i \le N$ and $a_i > a_{2i+1}$ for all $i$, $1 \le i < 2i+1 \le N$. For example, $T(3)$ is $2$, since the possible arrangements are $321$ and $312$ (a) Find $T(7)$ (b) If $K$ is the largest non-negative integer so that $2^K$ divides $T(2^n - 1)$, show that $K = 2^n - n - 1$. (c) Find the largest non-negative integer $K$ so that $2^K$ divides $T(2^n + 1)$

Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$. Find the minimum value and the maximum value of the product $abcd$.

Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)