Suppose that $x$, $y$, $z$ are real numbers satisfying \[x+y+z=12,\qquad\text{and}\qquad x^2+y^2+z^2=54.\] Prove that:[list](a) Each of the numbers $xy$, $yz$, $zx$ is at least $9$, but at most $25$. (b) One of the numbers $x$, $y$, $z$ is at most $3$, and another one is at least $5$.[/list]

Let be five nonzero complex numbers having the same absolute value and such that zero is equal to their sum, which is equal to the sum of their squares. Prove that the affixes of these numbers in the complex plane form a regular pentagon. [i]Daniel Jinga[/i]

In a $ 19\times 19$ board, a piece called [i]dragon[/i] moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board. The [i]draconian distance[/i] between two squares is defined as the least number of moves a dragon needs to move from one square to the other. Let $ C$ be a corner square, and $ V$ the square neighbor of $ C$ that has only a point in common with $ C$. Show that there exists a square $ X$ of the board, such that the draconian distance between $ C$ and $ X$ is greater than the draconian distance between $ C$ and $ V$.

The graph of $f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|$ is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.) $\textbf{(A) }$ the $y$-axis $\qquad \textbf{(B) }$ the line $x = 1$ $\qquad \textbf{(C) }$ the origin $\qquad \textbf{(D) }$ the point $\left(\dfrac12, 0\right)$ $\qquad \textbf{(E) }$ the point $(1,0)$

Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$

We write $1$ or $-1$ on each unit square of a $2007 \times 2007$ board. Find the number of writings such that for every square on the board the absolute value of the sum of numbers on the square is less then or equal to $1$.

In every cell of a $ 60 \times 60$ board is written a real number, whose absolute value is less or equal than $ 1$. The sum of all numbers on the board equals $ 600$. Prove that there is a $ 12 \times 12$ square in the board such that the absolute value of the sum of all numbers on it is less or equal than $ 24$.

Let $P(x), Q(x), R(x)$ be polynomials with integer coefficients, such that $P(x) = Q(x)R(x)$. Let's denote by $a$ and $b$ the largest absolute values of coefficients of $P, Q$ correspondingly. Does $b \le 2023a$ always hold? [i]Proposed by Dmytro Petrovsky[/i]

Let $x_1, ..., x_n$ $(n \geq 2)$ be real numbers from the interval $[1,2]$. Prove that $$|x_1-x_2|+...+|x_n-x_1| + \frac{1}{3} (|x_1-x_3|+...+|x_n-x_2|) \leq \frac{2}{3} (x_1+...+x_n)$$ and determine all cases of equality.

Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.

Find the median value of $m$ over all integers $m$ where $|m^2 + 8m - 65|$ is a perfect power. A perfect power is any integer at least $2$ which can be written as $a^b$, where $a$, $b$ are integers and $b \ge 2$. [i]Individual #9[/i]

Find the minimum value of \[|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}|\] for real numbers $x$.

Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

Let $a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that \[\left\vert a+b\sqrt{2}+c\sqrt{3}\right\vert > \frac{1}{10^{21}}.\]

The absolute value of the sum of the elements of a real orthogonal matrix is at most the order of the matrix.

$n$ integers are arranged along a circle in such a way that each number is equal to the absolute value of the difference of the two numbers following that number in clockwise direction. If the sum of all numbers is $278$, how many different values can $n$ take? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 139 \qquad\textbf{(E)}\ \text{None of the above} $

Every square of $1000 \times 1000$ board is colored black or white. It is known that exists one square $10 \times 10$ such that all squares inside it are black and one square $10 \times 10$ such that all squares inside are white. For every square $K$ $10 \times 10$ we define its power $m(K)$ as an absolute value of difference between number of white and black squares $1 \times 1$ in square $K$. Let $T$ be a square $10 \times 10$ which has minimum power among all squares $10 \times 10$ in this board. Determine maximal possible value of $m(T)$

Let $A\subset \{x|0\le x<1\}$ with the following properties: 1. $A$ has at least 4 members. 2. For all pairwise different $a,b,c,d\in A$, $ab+cd\in A$ holds. Prove: $A$ has infinetly many members.

Number $ 0$ is written on the board. Two players alternate writing signs and numbers to the right, where the first player always writes either $ \plus{}$ or $ \minus{}$ sign, while the second player writes one of the numbers $ 1, 2, ... , 1993$,writing each of these numbers exactly once. The game ends after $ 1993$ moves. Then the second player wins the score equal to the absolute value of the expression obtained thereby on the board. What largest score can he always win?

Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms: $$\sum_{i<j}|x_i -x_j |.$$

[b]a)[/b] Let $ z_1,z_2,z_3 $ be three complex numbers of same absolute value, and $ 0=z_1+z_2+z_3. $ Show that these represent the affixes of an equilateral triangle. [b]b)[/b] Find all subsets formed by roots of the same unity that have the property that any three elements of every such, doesn’t represent the vertices of an equilateral triangle.

Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that \[ \left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\] for every positive integer $n$.

Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. (You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.) [i]Ray Li[/i]

In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$. In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps?