Find functions $ f: \mathbb{N} \rightarrow \mathbb{N}$, such that $ f(x^2 \plus{} f(y)) \equal{} xf(x) \plus{} y$, for $ x,y \in \mathbb{N}$.

A cell phone plan costs $\$20$ each month, plus $5$¢ per text message sent, plus 10¢ for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay? $ \textbf{(A)}\$ 24.00\qquad\textbf{(B)}\$ 24.50\qquad\textbf{(C)}\$25.50\qquad\textbf{(D)}\$28.00\qquad\textbf{(E)}\$30.00 $

A parabola is inscribed in equilateral triangle $ABC$ of side length $1$ in the sense that $AC$ and $BC$ are tangent to the parabola at $A$ and $B$, respectively. Find the area between $AB$ and the parabola.

Is it possible to place 100 consecutive numbers around a circle in some order such that the product of each pair of adjacent numbers is always a perfect square? (Recall that a number is a perfect square if it is the square of an integer.)

[b]4.[/b] Let $F_{\epsilon} (0<\epsilon<1)$ denote the class of non-negative piecewise continuous functions defined on $[0,\infty)$ which satisfy the following condition: $f(x)f(y)\leq \epsilon^{\mid x-y\mid} (x,y \geq 0)$. Find the value of $s_{\epsilon}= \sup_{f\in F_{\epsilon}} \int_{0}^{\infty} f(x) dx$ [b](R. 5)[/b]

It is known that the graph of a quadratic trinomial $y = x^2 + px + q$ touches the graph of a straight line $y = 2x + p$. Prove that all such quadratic trinomials have the same minimum value. Find this smallest value.

Let $ p$ be an odd prime number and $ a_1,a_2,...,a_p$ and $ b_1,b_2,...,b_p$ two arbitrary permutations of the numbers $ 1,2,...,p$ . Show that the least positive residues modulo $ p$ of the numbers $ a_1b_1, a_2b_2,...,a_pb_p$ never form a permutation of the numbers $ 1,2,...,p$.

For any natural number, let's call the numbers formed from its digits and have the same "digit" arrangement with the initial number as the "partial numbers". For example, the partial numbers of $149$ are ${1, 4, 9, 14,19, 49, 149},$ and the partial numbers of $313$ are ${3, 1, 31,33, 13, 313}.$ Find all natural numbers whose partial numbers are all prime. Justify your opinion.

In a badminton tournament there were 16 participants. Each pair of participants played at most one game and there were no draws. After the tournament it turned out that each participant has won a different number of games. Prove that each participant has lost a different number of games.

Let $m$ be a positive integer and $\{a_n\}_{n\geq 0}$ be a sequence given by $a_0 = a \in \mathbb N$, and \[ a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + m & \textrm{ otherwise. } \end{cases} \] Find all values of $a$ such that the sequence is periodical (starting from the beginning).

Evaluate the sum $\frac{3!+4!}{2(1!+2!)}+\frac{4!+5!}{3(2!+3!)}+\cdots+\frac{12!+13!}{11(10!+11!)}$

Alice and Bob play the following game: They start with non-empty piles of coins. Taking turns, with Alice playing first, each player choose a pile with an even number of coins and moves half of the coins of this pile to the other pile. The game ends if a player cannot move, in which case the other player wins. Determine all pairs $(a,b)$ of positive integers such that if initially the two piles have $a$ and $b$ coins respectively, then Bob has a winning strategy. Proposed by Dimitris Christophides, Cyprus

Find the three-digit number that has the greatest number of different divisors.

A simple graph has $N{}$ vertices and less than $3(N-1)/2$ edges. Prove that its vertices can be divided into two non-empty groups so that each vertex has at most one neighbor in the group it doesn't belong to.

In a semicircle with center $O$, let $C$ be a point on the diameter $AB$ different from $A, B$ and $O.$ Draw through $C$ two rays such that the angles that these rays form with the diameter $AB$ are equal and that they intersect at the semicircle at $D$ and at $E$. The line perpendicular to $CD$ through $D$ intersects the semicircle at $K.$ Prove that if $D\neq E,$ then $KE$ is parallel to $AB.$

Let $k$, $m$ and $n$ be integers such that $1<n \leq m-1 \leq k$. Find maximum size of subset $S$ of set $\{1,2,...,k\}$ such that sum of any $n$ different elements from $S$ is not: $a)$ equal to $m$, $b)$ exceeding $m$

A $10 \times 10$ table consists of $100$ unit cells. A [i]block[/i] is a $2 \times 2$ square consisting of $4$ unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$ ) but no $n -1$ blocks of $C$ cover the table. Find the largest possible value of $n$.

[b](i)[/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 3-clique (3 nodes joined pairwise by edges). [b](ii)[/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 4-clique (4 nodes joined pairwise by edges).

Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.

Mr. Squash bought a large parking lot in Utah, which has an area of $600$ square meters. A car needs $6$ square meters of parking space while a bus needs $30$ square meters of parking space. Mr. Squash charges $\$2.50$ per car and $\$7.50$ per bus, but Mr. Squash can only handle at most $60$ vehicles at a time. Find the ordered pair $(a,b)$ where $a$ is the number of cars and $b$ is the number of buses that maximizes the amount of money Mr. Squash makes. [i]Proposed by Nathan Cho[/i]

In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? [asy] unitsize(150); pair A,B,C,D,E,F,G,H,J,K; A=(1,0); B=(0,0); C=(0,1); D=(1,1); draw(A--B--C--D--A); E=(2-sqrt(3),0); F=(2-sqrt(3),1); draw(E--F); G=(1,sqrt(3)/2); H=(2.5-sqrt(3),1); K=(2-sqrt(3),1-sqrt(3)/2); J=(0.5,0); draw(G--H--K--J--G); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$E$",E,S); label("$F$",F,N); label("$G$",G,E); label("$H$",H,N); label("$K$",K,W); label("$J$",J,S); [/asy] $ \textbf{(A) }\dfrac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\dfrac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\dfrac{\sqrt{3}}{6}\qquad\textbf{(E) }1-\dfrac{\sqrt{2}}{2} $

$n$ assistants start simultaneously from one vertex of a cube-shaped planet with edge length $1$. Each assistant moves along the edges of the cube at a constant speed of $2, 4, 8, \cdots, 2^n$, and can only change their direction at the vertices of the cube. The assistants can pass through each other at the vertices, but if they collide at any point that is not a vertex, they will explode. Determine the maximum possible value of $n$ such that the assistants can move infinitely without any collisions.

Let $ABC$ be an obtuse triangle with $AB = AC$, and let $\Gamma$ be the circle that is tangent to $AB$ at $B$ and to $AC$ at $C$. Let $D$ be the point on $\Gamma$ furthest from $A$ such that $AD$ is perpendicular to $BC$. Point $E$ is the intersection of $AB$ and $DC$, and point $F$ lies on line $AB$ such that $BC = BF$ and $B$ lies on segment $AF$. Finally, let $P$ be the intersection of lines $AC$ and $DB$. Show that $PE = PF$.

Let $\omega$ be the incircle of a triangle $ABC$. The line passing though the incenter $I$ and parallel to $BC$ meets $\omega$ at $A_b$ and $A_c$ ($A_b$ lies in the same semi plane with respect to $AI$ as $B$). The lines $BA_b$ and $CA_c$ meet at $A_1$. The points $B_1$ and $C_1$ are defined similarly. prove that $AA_1,BB_1,CC_1$ concur.

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$