For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

Find the area of the region of the $xy$-plane defined by the inequality $|x|+|y|+|x+y| \le 1$.

Find all real-valued functions on the positive integers such that $f(x + 1019) = f(x)$ for all $x$, and $f(xy) = f(x) f(y)$ for all $x,y$.

Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet? [asy] unitsize(0.7cm); path p1 = (0,0)--(15,0)--(15,10)--(0,10)--cycle; fill(p1,lightgray); draw(p1); for (int i = 1; i <= 8; i += 7) { for (int j = 1; j <= 7; j += 3 ) { path p2 = (i,j)--(i+6,j)--(i+6,j+2)--(i,j+2)--cycle; draw(p2); fill(p2,white); } } draw((0,8)--(1,8),Arrows); label("1",(0.5,8),S); draw((7,8)--(8,8),Arrows); label("1",(7.5,8),S); draw((14,8)--(15,8),Arrows); label("1",(14.5,8),S); draw((11,0)--(11,1),Arrows); label("1",(11,0.5),W); draw((11,3)--(11,4),Arrows); label("1",(11,3.5),W); draw((11,6)--(11,7),Arrows); label("1",(11,6.5),W); draw((11,9)--(11,10),Arrows); label("1",(11,9.5),W); label("6",(4,1),N); label("2",(1,2),E); [/asy] $\textbf{(A) }72 \qquad \textbf{(B) }78 \qquad \textbf{(C) }90 \qquad \textbf{(D) }120 \qquad \textbf{(E) }150 $

Show that any arithmetic progression where the first term and the common difference are non-zero natural numbers contains an infinite number of composite terms. *A number is composite if it is not prime.

Let $ABCD$ be a rectangle and let $E \in (BD)$ such that $m( \angle DAE) =15^o$. Let $F \in AB$ such that $EF \perp AB$. It is known that $EF=\frac12 AB$ and $AD = a$. Find the measure of the angle $\angle EAC$ and the length of the segment $(EC)$.

Let $ABCD$ be an inscribed quadrilateral in a circle $c(O,R)$ (of circle $O$ and radius $R$). With centers the vertices $A,B,C,D$, we consider the circles $C_{A},C_{B},C_{C},C_{D}$ respectively, that do not intersect to each other . Circle $C_{A}$ intersects the sides of the quadrilateral at points $A_{1} , A_{2}$ , circle $C_{B}$ intersects the sides of the quadrilateral at points $B_{1} , B_{2}$ , circle $C_{C}$ at points $C_{1} , C_{2}$ and circle $C_{D}$ at points $C_{1} , C_{2}$ . Prove that the quadrilateral defined by lines $A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}$ is cyclic.

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that \[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]

Given an acute triangle $ABC$. Points $B'$ and $C'$ are symmetrical, respectively, to vertices $B$ and $ C$ wrt straight lines $AC$ and $AB$. Let $P$ be the intersection point of the circumcircles of triangles $ABB'$ and $ACC'$, different from $A$. Prove that the center of the circumcircle of triangle $ABC$ lies on line $PA$.

Let $p,q$ be prime numbers such that $n^{3pq}-n$ is a multiple of $3pq$ for [b]all[/b] positive integers $n$. Find the least possible value of $p+q$.

Prove that there are infinitely many positive integers $m$ for which there exists consecutive odd positive integers $p_m<q_m$ such that $p_m^2+p_mq_m+q_m^2$ and $p_m^2+m\cdot p_mq_m+q_m^2$ are both perfect squares. If $m_1, m_2$ are two positive integers satisfying this condition, then we have $p_{m_1}\neq p_{m_2}$

Let $N>1$ be an integer. We are adding all remainders when we divide $N$ by all positive integers less than $N$. If this sum is less than $N$, find all possible values of $N$.

Initially, the number $1$ is written on the blackboard. At each step, the number on the blackboard is erased and another is written, which is obtained by applying any of the following operations: Operation A: Multiply the number on the board with $\frac12$. Operation B: Subtract the number on the board from $1$. For example, if the number $\frac38$ is on the board, it can be replaced by $\frac12 \frac38=\frac{3}{16}$ or by $1-\frac38=\frac58$ . Give a sequence of steps after which the number on the board is $\frac{2009}{2^{20009}}$ .

Determine the smallest positive integer $ n$ such that there exists positive integers $ a_1,a_2,\cdots,a_n$, that smaller than or equal to $ 15$ and are not necessarily distinct, such that the last four digits of the sum, \[ a_1!\plus{}a_2!\plus{}\cdots\plus{}a_n!\] Is $ 2001$.

Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.

In a triangle $ABC$ with $AC\ne BC$, $M$ is the midpoint of $AB$ and $\angle A=\alpha$, $\angle B=\beta$, $\angle ACM=\varphi$ and $\angle BSM=\Psi$. Prove that $$\frac{\sin\alpha\sin\beta}{\sin(\alpha-\beta)}=\frac{\sin\varphi\sin\Psi}{\sin(\varphi-\Psi)}.$$

Prove that for all n=3,4,5.... there excist odd x,y such $2^n=x^2 + 7y^2$ .

A convex quadrilateral is inscribed in a circle of radius $1$. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than $2.$

Which of the following integers cannot be written as the sum of four consecutive odd integers? $\textbf{(A)}\text{ 16}\qquad\textbf{(B)}\text{ 40}\qquad\textbf{(C)}\text{ 72}\qquad\textbf{(D)}\text{ 100}\qquad\textbf{(E)}\text{ 200}$

Find the minimum value of \[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\] where $x,y,z>1$ are reals.

Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$.

Let $ABC$ be a triangle with $BC=7,AB=5$, and $AC=8$. Let $M,N$ be the midpoints of sides $AC,AB$ respectively, and let $O$ be the circumcenter of $ABC$. Let $BO, CO$ meet $AC, AB$ at $P$ and $Q$, respectively. If $MN$ meets $PQ$ at $R$ and $OR$ meets $BC$ at $S$, then the value of $OS^2$ can be written in the form $\frac{m}{n}$ where $m,n$ are relatively prime positive integers. Find $100m+n$. [i]Proposed by Vincent Huang[/i]

In the fourth annual Swirled Series, the Oakland Alphas are playing the San Francisco Gammas. The first game is played in San Francisco and succeeding games alternate in location. San Francisco has a $50\%$ chance of winning their home games, while Oakland has a probability of $60\%$ of winning at home. Normally, the series will stretch on forever until one team gets a three game lead, in which case they are declared the winners. However, after each game in San Francisco there is a $50\%$ chance of an earthquake, which will cause the series to end with the team that has won more games declared the winner. What is the probability that the Gammas will win?

A coast artillery gun can fire at every angle of elevation between $0^{\circ}$ and $90^{\circ}$ in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant ($=v_0 $), determine the set $H$ of points in the plane and above the horizontal which can be hit.