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 $10$?

Ten bowls are in a circle. They will go clockwise - starting somewhere filled with $1, 2, 3, ..., 9$ or $10$ marbles. You can have two choices in every move . Add a marble to neighboring shells or from two neighboring shells - if both of them are not empty - remove one marble each. Can you achieve that after finally many moves in each bowl exactly $2011$ marbles lying?

Find all ordered triples $(a,b,c)$ of real numbers such that \[\begin{cases} a+b=c,\\ a^2+b^2=c^2-c-6,\\ a^3+b^3 = c^3-2c^2-5c. \\ \end{cases}\] [i]Proposed by Evan Fang

Let $D$ be a point on side $[BC]$ of triangle $ABC$ where $|BC|=11$ and $|BD|=8$. The circle passing through the points $C$ and $D$ touches $AB$ at $E$. Let $P$ be a point on the line which is passing through $B$ and is perpendicular to $DE$. If $|PE|=7$, then what is $|DP|$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None of above} $

Let $ABC$ be a triangle with $AB> AC$. Let $D$ be a point on $AB$ such that $DB = DC$ and $M$ the middle of $AC$. The parallel to $BC$ passing through $D$ intersects the line $BM$ in $K$. Show that $\angle KCD = \angle DAC$.

Square $ AIME$ has sides of length $ 10$ units. Isosceles triangle $ GEM$ has base $ EM$, and the area common to triangle $ GEM$ and square $ AIME$ is $ 80$ square units. Find the length of the altitude to $ EM$ in $ \triangle GEM$.

Let $\alpha$, $\beta$ and $\gamma$ be the roots of the polynomial $2023x^3-2023x^2-1$. Find $$\frac{1}{\alpha^3}+\frac{1}{\beta^3}+\frac{1}{\gamma^3}$$. [i]Proposed by Andy Xu[/i]

Let $P$ an interior point of an equilateral triangle $ABC$. Prove that there exists triangle with sides $PA , PB , PC$ . Babis