Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that: (1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$ (2) If $m,n \in S$ , then $mn\in S$.

Let $ABCDEF$ be a convex hexagon. The diagonal $AC$ is cut by $BF$ and $BD$ at points $P$ and $Q$, respectively. The diagonal $CE$ is cut by $DB$ and $DF$ at points $R$ and $S$, respectively. The diagonal $EA$ is cut by $FD$ and $FB$ at points $T$ and $U$, respectively. It is known that each of the seven triangles $APB, PBQ, QBC, CRD, DRS, DSE$ and $AUF$ has area $1$. Find the area of the hexagon $ABCDEF$.

Three positive real numbers $a,b,c$ are such that $a^2+5b^2+4c^2-4ab-4bc=0$. Can $a,b,c$ be the lengths of te sides of a triangle? Justify your answer.

Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively. If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .

In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If $AG=2$, $GF=13$, $FC=1$, and $HJ=7$, then $DE$ equals [asy] size(200); defaultpen(fontsize(10)); real r=sqrt(22); pair B=origin, A=16*dir(60), C=(16,0), D=(10-r,0), E=(10+r,0), F=C+1*dir(120), G=C+14*dir(120), H=13*dir(60), J=6*dir(60), O=circumcenter(G,H,J); dot(A^^B^^C^^D^^E^^F^^G^^H^^J); draw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7)); label("$A$", A, N); label("$B$", B, dir(210)); label("$C$", C, dir(330)); label("$D$", D, SW); label("$E$", E, SE); label("$F$", F, dir(170)); label("$G$", G, dir(250)); label("$H$", H, SE); label("$J$", J, dir(0)); label("2", A--G, dir(30)); label("13", F--G, dir(180+30)); label("1", F--C, dir(30)); label("7", H--J, dir(-30));[/asy] $\textbf {(A) } 2\sqrt{22} \qquad \textbf {(B) } 7\sqrt{3} \qquad \textbf {(C) } 9 \qquad \textbf {(D) } 10 \qquad \textbf {(E) } 13$

There is a billiard table in shape of rectangle $2 \times 1$, with pockets at its corners and at midpoints of its two largest sizes. Find the minimal number of balls one has to place on the table interior so that any pocket is on a straight line with some two balls. (Assume that pockets and balls are points). [i](4 points)[/i]

Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $ABC$. Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C$. The perpendicular from $L$ to $BC$ meets $AP$ at point $Q$. Prove that $Q$ lies on the medial line of triangle $LKP$.

Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

Let $A$ denote the set of all the positive integer divisors of $30.$ For each nonempty subset $s \subseteq A,$ define $p(s)$ to be the product of the elements in $s.$ Finally, let $B$ denote the set of all possible remainders when $p(s)$ is divided by $30.$ How many (distinct) elements are in $B?$

Prove that for all $n\in\mathbb{Z}^+$, we have \[ \sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3} \]

Madoka chooses $4$ random numbers $a, b, c, d$ between $0$ and $1$. She notices that $a+b+c = 1$. If the probability that $d > a, b, c$ can be written in simplest form as $\frac{m}{n}$, find $m + n$.

Let $P(x)$ be a polynomial with integer coefficients. Show that if $Q(x) = P(x) +12$ has at least six distinct integer roots, then $P(x)$ has no integer roots.

A sequence of primes $a_n$ is defined as follows: $a_1 = 2$, and, for all $n \geq 2$,$ a_n$ is the largest prime divisor of $a_1a_2...a_{n-1} + 1$. Prove that $a_n \neq 5$ for all n. I'm presuming it must involve proving it's never equal to 0 mod 5, but I don't know what to do. Thanks

We are given $n>1$ piles of coins. There are two different types of coins: real and fake coins; they all look alike, but coins of the same type have the same mass, while the coins from different types have different masses. Coins that belong to the same pile are of the same type. We know the mass of real coin. Find the minimal number of weightings on digital scale that we need in order to conclude: which piles consists of which type of coins and also the mass of fake coin. (We assume that every pile consists from infinite number of coins.)

Given a sequence {$a_n$} such that $a_1=2$ and for all positive integer $n\geq 2$ $a_{n+1}=\frac{a_n^4+9}{16a_n}$. Prove that $\frac {4}{5}<a_n<\frac {5}{4}$

Points $P_1, P_2, P_3,$ and $P_4$ are $(0,0), (10, 20), (5, 15),$ and $(12, -6)$, respectively. For what point $P \in \mathbb{R}^2$ is the sum of the distances from $P$ to the other $4$ points minimal?

Let $ ABCD$ be an isosceles trapezoid with $ AD \parallel BC$. Its diagonals $ AC$ and $ BD$ intersect at point $ M$. Points $ X$ and $ Y$ on the segment $ AB$ are such that $ AX \equal{} AM$, $ BY \equal{} BM$. Let $ Z$ be the midpoint of $ XY$ and $ N$ is the point of intersection of the segments $ XD$ and $ YC$. Prove that the line $ ZN$ is parallel to the bases of the trapezoid. [i]Author: A. Akopyan, A. Myakishev[/i]

Find the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$.

A function $\psi \colon {\mathbb Z} \to {\mathbb Z}$ is said to be [i]zero-requiem[/i] if for any positive integer $n$ and any integers $a_1$, $\ldots$, $a_n$ (not necessarily distinct), the sums $a_1 + a_2 + \dots + a_n$ and $\psi(a_1) + \psi(a_2) + \dots + \psi(a_n)$ are not both zero. Let $f$ and $g$ be two zero-requiem functions for which $f \circ g$ and $g \circ f$ are both the identity function (that is, $f$ and $g$ are mutually inverse bijections). Given that $f+g$ is [i]not[/i] a zero-requiem function, prove that $f \circ f$ and $g \circ g$ are both zero-requiem. [i]Sutanay Bhattacharya[/i]

Find the sum of all the possible values of xy such that x and y are positive integers satisfying $(x^2 + 1)(y^2 + 1) + 2(x -y)(1 - xy) = 4(1 + xy) + 140$.

$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?

For each $ c\in\mathbb C$, let $ f_c(z,0)\equal{}z$, and $ f_c(z,n)\equal{}f_c(z,n\minus{}1)^2\plus{}c$ for $ n\geq1$. a) Prove that if $ |c|\leq\frac14$ then there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is bounded. b) Prove that if $ c>\frac14$ is a real number there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is unbounded.

The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that $f(x+1)\leq f(2x+1)$ and $f(3x+1)\geq f(6x+1)$ for $\forall$ $x\in \mathbb{R}$. If $f(3)=2$, prove that there exist at least 2013 distinct values of $x$, for which $f(x)=2$.

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]