$n$ is a integer and $a_1, a_2, \ldots, a_n\in[-1,1]$ are real numbers with $ \sum_{i=1}^{n}a_{i}=0$ ,try to find the maximum value of $$ \sum_{1\leq i , j \leq n , i\ne j}|a_{i}-a^2_j|$$

The real numbers $a,b,c$ satisfy the condition: for all $x$, such that for $ -1 \le x \le 1$, the inequality $$| ax^2 + bx + c | \le 1$$ is held. Prove that for the same $x$ , $$| cx^2 + bx + a | \le 2$$

The walls of a room are in the shape of a triangle $ABC$ with $\angle ABC = 90^\circ$, $\angle BAC = 60^\circ$, and $AB=6$. Chong stands at the midpoint of $BC$ and rolls a ball toward $AB$. Suppose that the ball bounces off $AB$, then $AC$, then returns exactly to Chong. Find the length of the path of the ball.

Let $D$ and $D'$ be two lines with the equations \[\frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \quad \text{and} \quad \frac{x+1}{2} = \frac{y+2}{4} = \frac{z-1}{3}.\] Find the length of their common perpendicular.

A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]

Let $a$,$b$,$c$ be real numbers. Let $z_{1}$,$z_{2}$,$z_{3}$ be complex numbers such that $|z_{k}|=1$ $(k=1,2,3)$ $~$ and $~$ $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1$ Find $|az_{1}+bz_{2}+cz_{3}|$.

Julia and James pick a random integer between $1$ and $10$, inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

A graph $G$ with $n$ vertex is called [i]good [/i] if every vertex could be labelled with distinct positive integers which are less than or equal $\lfloor \frac{n^2}{4} \rfloor$ such that there exists a set of nonnegative integers $D$ with the following property: there exists an edge between $2$ vertices if and only if the difference of their labels is in $D$. Show that there exists a positive integer $N$ such that for every $n \ge N$, there exist a not-good graph with $n$ vertices.

Let $n$ be a natural number. Determine the minimal number of equilateral triangles of side $1$ to cover the surface of an equilateral triangle of side $n +\frac{1}{2n}$.

Among 6 coins one is counterfeit (its weight differs from that real one and neither weights is known). Using scales that show the total weight of coins placed on the cup, find the counterfeit coin in 3 weighings. [i](4 points)[/i]

The diagonals of a rectangle $ABCD$ meet at point $E$. A circle centered at $E$ lies inside the rectangle. Let $CF$, $DG$, $AH$ be the tangents to this circle from $C$, $D$, $A$; let $CF$ meet $DG$ at point $I$, $EI$ meet $AD$ at point $J$, and $AH$ meet $CF$ at point $L$. Prove that $LJ$ is perpendicular to $AD$.

A circle of radius $a$ is revolved through $180^{\circ}$ about a line in its plane, distant $b$ from the center of the circle, where $b>a$. For what value of the ratio $\frac{b}{a}$ does the center of gravity of the solid thus generated lie on the surface of the solid?

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

A certain organization has $n$ members, and it has $n\plus{}1$ three-member committees, no two of which have identical member-ship. Prove that there are two committees which share exactly one member.

Let $\varphi$ denote the Euler phi-function. Prove that for every positive integer $n$ $$2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).$$

Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. [i]Proposed by Michal Rolinek, Czech Republic[/i]

Given a line segment $PQ$ with endpoints on the parabola $y=x^2$ such that the area bounded by $PQ$ and the parabola always equal to $\frac 43.$ Find the equation of the locus of the midpoint $M$.

Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$, let $k_n$ be the least positive integer satisfying: \[\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.\] Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.

A natural number $n$ is written on the board. Ben plays a game as follows: in every step, he deletes the number written on the board, and writes either the number which is three greater or two less than the number he has deleted. Is it possible that for every value of $n$, at some time, he will get to the number $2020$?

Prove that $2^{n - 1}$ divides $n!$ if and only if $n = 2^{k - 1}$ for some positive integer $k$.

For each positive integer $n$, let $d(n)$ be the number of positive integer divisors of $n$. Prove that for all pairs of positive integers $(a,b)$ we have that: \[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \] and determine all pairs of positive integers $(a,b)$ where we have equality case.

John has a standard four-sided die. Each roll, he gains points equal to the value of the roll multiplied by the number of times he has now rolled that number; for example, if his first rolls were $3,3,2,3$, he would have $3+6+2+9=20$ points. Find the expected number of points John will have after rolling the die 25 times.

Suppose $f:\mathbb{N}\rightarrow \mathbb{N}$ is a function such that $$f^n(n) = 2n$$ for all $n\in \mathbb{N}$. Must $f(n) = n+1$ for all $n$?

Find all composite numbers $n$ such that for each decomposition of $n=xy$, $x+y$ be a power of $2$.