Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that \[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\] for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that \[a_{k+2011}=a_{k+2011^{2011}}.\]

Let $a,b,c,d$ real numbers such that: $$ a+b+c+d=0 \text{ and } a^2+b^2+c^2+d^2 = 12 $$ Find the minimum and maximum possible values for $abcd$, and determine for which values of $a,b,c,d$ the minimum and maximum are attained.

Determine the number of real roots of the equation \[x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x +\frac{5}{2}= 0.\]

Let $a, b$ and $k$ be positive integers with $k> 1$ such that $lcm (a, b) + gcd (a, b) = k (a + b)$. Prove that $a + b \geq 4k$

Call an ordered pair $(a, b)$ of positive integers [i]fantastic [/i] if and only if $a, b \le 10^4$ and $$gcd(a \cdot n! - 1, a \cdot (n + 1)! + b) > 1$$ for infinitely many positive integers $n$. Find the sum of $a + b$ across all fantastic pairs $(a, b)$.

Let $\omega$ be a circle with centre $O$. Let $\gamma$ be another circle passing through $O$ and intersecting $\omega$ at points $A$ and $B$. $A$ diameter $CD$ of $\omega$ intersects $\gamma$ at a point $P$ different from $O$. Prove that $\angle APC= \angle BPD$

How many rooks can be placed in an $n\times n$ chessboard such that each rook is threatened by at most $2k$ rooks? (15 points) [i]Proposed by Mostafa Einollah zadeh[/i]

Suppose that $a,b,c$ are real numbers such that $\left|ax^2+bx+c\right|\le1$ for $-1\le x\le1$. Prove that $\left|cx^2+bx+a\right|\le2$ for $-1\le x\le1$.

In triangle $ABC$, $AB \ne AC$. Let $AF$, $AP$ and $AT$ be the median, angle bisector and altitude from vertex $A$, with $F, P$ and $T$ on $BG$ or its extension. (a) Prove that $P$ always lies between$ F$ and $T$. (b) Prove that $\angle FAP < \angle PAT$ if $ABC$ is an acute triangle.

Let $u_n$ be the $n^\text{th}$ term of the sequence \[1,\,\,\,\,\,\,2,\,\,\,\,\,\,5,\,\,\,\,\,\,6,\,\,\,\,\,\,9,\,\,\,\,\,\,12,\,\,\,\,\,\,13,\,\,\,\,\,\,16,\,\,\,\,\,\,19,\,\,\,\,\,\,22,\,\,\,\,\,\,23,\ldots,\] where the first term is the smallest positive integer that is $1$ more than a multiple of $3$, the next two terms are the next two smallest positive integers that are each two more than a multiple of $3$, the next three terms are the next three smallest positive integers that are each three more than a multiple of $3$, the next four terms are the next four smallest positive integers that are each four more than a multiple of $3$, and so on: \[\underbrace{1}_{1\text{ term}},\,\,\,\,\,\,\underbrace{2,\,\,\,\,\,\,5}_{2\text{ terms}} ,\,\,\,\,\,\,\underbrace{6,\,\,\,\,\,\,9,\,\,\,\,\,\,12}_{3\text{ terms}},\,\,\,\,\,\,\underbrace{13,\,\,\,\,\,\,16,\,\,\,\,\,\,19,\,\,\,\,\,\,22}_{4\text{ terms}},\,\,\,\,\,\,\underbrace{23,\ldots}_{5\text{ terms}},\,\,\,\,\,\,\ldots.\] Determine $u_{2008}$.

Can an arc of a parabola inside a circle of radius $1$ have a length greater than $4$?

Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.

Let us choose arbitrarily $n$ vertices of a regular $2n$-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.

Consider $\triangle MAB$ with a right angle at $A$ and circumcircle $\omega$. Take any chord $CD$ perpendicular to $AB$ such that $A, C, B, D, M$ lie on $\omega$ in this order. Let $AC$ and $MD$ intersect at point $E$, and let $O$ be the circumcenter of $\triangle EMC$. Show that if $J$ is the intersection of $BC$ and $OM$, then $JB = JM$. [i](Proposed by Matthew Kung Wei Sheng and Ivan Chan Kai Chin)[/i]

Let $ f: Z \to Z$ be such that $ f(1) \equal{} 1, f(2) \equal{} 20, f(\minus{}4) \equal{} \minus{}4$ and $ f(x\plus{}y) \equal{} f(x) \plus{}f(y)\plus{}axy(x\plus{}y)\plus{}bxy\plus{}c(x\plus{}y)\plus{}4 \forall x,y \in Z$, where $ a,b,c$ are constants. (a) Find a formula for $ f(x)$, where $ x$ is any integer. (b) If $ f(x) \geq mx^2\plus{}(5m\plus{}1)x\plus{}4m$ for all non-negative integers $ x$, find the greatest possible value of $ m$.

Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.

There are $5$ students on a team for a math competition. The math competition has $5$ subject tests. Each student on the team must choose $2$ distinct tests, and each test must be taken by exactly two people. In how many ways can this be done?

The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111$, $R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=\frac{R_{24}}{R_4}$ is an integer whose base-ten representation is a sequence containing only ones and zeros. The number of zeros in $Q$ is $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 15 $

Let us call a pentagon curved, if all its sides are arcs of some circles. Are there exist a curved pentagon $P$ and a point $A$ on its boundary so that any straight line passing through $A$ divides perimeter of $P$ into two parts of the same length? [i](7 points)[/i]

Compute $\displaystyle\sum_{k=1}^\infty \dfrac{k^4}{k!}$.

Let $a,b$ be real numbers such that the equation $$x^3+\sqrt{3}(a-1)x^2-6ax+b=0$$has three real roots. Prove that $|b|\leq |a+1|^3$. >Clarification: $|x|$ indicates the absolute value of $x$. For example, $|5|=5$; $|-1.23|=1.23$; etc

Find the largest number among the following numbers: $ \textbf{(A) }\tan47^{\circ}+\cos47^{\circ}\qquad\textbf{(B) }\cot 47^{\circ}+\sqrt{2}\sin 47^{\circ}\qquad\textbf{(C) }\sqrt{2}\cos47^{\circ}+\sin47^{\circ}\qquad\textbf{(D) }\tan47^{\circ}+\cot47^{\circ}\qquad\textbf{(E) }\cos47^{\circ}+\sqrt{2}\sin47^{\circ} $

We call a tiling of an $m \times n$ rectangle with corners (see figure below) "regular" if there is no sub-rectangle which is tiled with corners. Prove that if for some $m$ and $n$ there exists a "regular" tiling of the $m \times n$ rectangular then there exists a "regular" tiling also for the $2m \times 2n $ rectangle.

Let $x_1,x_2,\ldots,x_n$ be positive reals. Prove that \[ \frac 1{1+x_1} + \frac 1{1+x_1+x_2} + \cdots + \frac 1{1+x_1+\cdots + x_n} < \sqrt { \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n}} . \] [i]Bogdan Enescu[/i]

In stage $0$ the numbers are written: $1 , 1$. In stage $1$ the sum of the numbers is inserted: $1, 2, 1$. In stage $2$, between each pair of numbers from the previous stage, the sum of them is inserted: $1, 3, 2, 3, 1$. One more stage: $1, 4, 3, 5, 2, 5, 3, 4, 1$. How many numbers are there in stage $10$? What is the sum of all the numbers in stage $10$?