Given a positive integer $n$. An inversion of a permutation is the amount of pairs $(i,j)$ such that $i<j$ and the $i$-th number is smaller than $j$-th number in the permutation. Prove that for every positive integer $k \leq n$ there exist exactly $\frac{n!}{k}$ permutations in which the inversion is divisible by $k$.

Let $n\geq1$ be an integer. We have two sequences, each of $n$ positive real numbers $a_1,a_2,\ldots ,a_n$ and $b_1,b_2,\ldots ,b_n$ such that $a_1+a_2+\ldots +a_n=1$ and $ b_1+b_2+\ldots +b_n=1$. Find the smallest possible value that the sum can take $$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+\ldots +\frac{a_n^2}{a_n +b_n}.$$

Let $ a,b,c$ positive reals such that $ ab \plus{} bc \plus{} ca \equal{} 3$, show that: $ \displaystyle a^2 \plus{} b^2 \plus{} c^2 \plus{} 3 \ge \frac {a(3 \plus{} bc)^2}{(c \plus{} b)(b^2 \plus{} 3)} \plus{} \frac {b(3 \plus{} ca)^2}{(a \plus{} c)(c^2 \plus{} 3)} \plus{} \frac {c(3 \plus{} ab)^2}{(b \plus{} a)(a^2 \plus{} 3)}$ ([i]Anass BenTaleb, Ali Ben Bari High School - Taza,Morocco[/i])

Let $O$ be a point inside a triangle $ABC$ and let the lines $AO,BO$, and $CO$ meet sides $BC,CA$, and $AB$ at points $A_1,B_1$, and $C_1$, respectively. If $AA_1$ is the longest among the segments $AA_1,BB_1,CC_1$, prove that $$OA_1+OB_1+OC_1\le AA_1.$$

Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation \[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\] holds.

Prove that for every $\varepsilon >0$ there exists a positive integer $n$ and there are positive numbers $a_1, \ldots, a_n$ such that for every $\varepsilon < x < 2\pi - \varepsilon$ we have $$\sum_{k=1}^n a_k\cos kx < -\frac{1}{\varepsilon}\left| \sum_{k=1}^n a_k\sin kx\right|$$.

The non-negative real numbers $x, y, z$ satisfy the equality $x + y + z = 1$. Determine the highest possible value of the expression $E (x, y, z) = (x + 2y + 3z) (6x +3y + 2z)$.

Mark six points in a plane so that any three of them are vertices of a nondegenerate isosceles triangle.

In triangle $ABC$, $M$ is the intersection point of the medians, $O$ is the center of the inscribed circle. Prove that if the line $OM$ is parallel to the side $BC$, then the point $O$ is equidistant from the sides $AB$ and $AC$.

A pile of cards, numbered with $1$, $2$, ..., $n$, is being shuffled. Afterwards, the following operation is repeatedly performed: If the uppermost card of the pile has the number $k$, then we reverse the order of the $k$ uppermost cards. Prove that, after finitely many executions of this operation, the card with the number $1$ will become the uppermost card of the pile.

A positive integer $n$ is [i]friendly[/i] if the difference of each pair of neighbouring digits of $n$, written in base $10$, is exactly $1$. [i]For example, 6787 is friendly, but 211 and 901 are not.[/i] Find all odd natural numbers $m$ for which there exists a friendly integer divisible by $64m$.

Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$. \[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \]

Let $n>1$ be an even positive integer. An $2n \times 2n$ grid of unit squares is given, and it is partitioned into $n^2$ contiguous $2 \times 2$ blocks of unit squares. A subset $S$ of the unit squares satisfies the following properties: (i) For any pair of squares $A,B$ in $S$, there is a sequence of squares in $S$ that starts with $A$, ends with $B$, and has any two consecutive elements sharing a side; and (ii) In each of the $2 \times 2$ blocks of squares, at least one of the four squares is in $S$. An example for $n=2$ is shown below, with the squares of $S$ shaded and the four $2 \times 2$ blocks of squares outlined in bold. [asy] size(2.5cm); fill((0,0)--(4,0)--(4,1)--(0,1)--cycle,mediumgrey); fill((0,0)--(0,4)--(1,4)--(1,0)--cycle,mediumgrey); fill((0,3)--(4,3)--(4,4)--(0,4)--cycle,mediumgrey); fill((3,0)--(3,4)--(4,4)--(4,0)--cycle,mediumgrey); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw((1,0)--(1,4)); draw((2,0)--(2,4),linewidth(1)); draw((3,0)--(3,4)); draw((0,1)--(4,1)); draw((0,2)--(4,2),linewidth(1)); draw((0,3)--(4,3)); [/asy] In terms of $n$, what is the minimum possible number of elements in $S$?

Let $k \geq 2$ and $n_1, n_2, . . . , n_k \geq 1$ natural numbers having the property $n_2 | 2^{n_1} - 1, n_3 | 2^{n_2} -1 , \cdots, n_k | 2^{n_k-1}-1$, and $n_1 | 2^{n_k} - 1$. Show that $n_1 = n_2 = \cdots = n_k = 1.$

Problem 5. Consider the integer number n>2. Let a_1,a_2,…,a_n and b_1,b_2,…,b_n be two permutations of 0,1,2,…,n-1. Prove that there exist some i≠j such that: n|a_i b_i-a_j b_j [color=#00f]Moved to HSO. ~ oVlad[/color]

In a cyclic quadrilateral $ABCD$ the line passing through the midpoint of $AB$ and the intersection point of the diagonals is perpendicular to $CD$. Prove that either the sides $AB$ and $CD$ are parallel or the diagonals are perpendicular.

Find all integer triples $(a,m,n)$ such that $a^m+1|a^n+203$ where $a,m>1$. [i]Chen Yonggao[/i]

Find the greatest value of the expression \[ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} \] where $x$, $y$, $z$ are nonnegative real numbers such that $x+y+z=1$.

Construct a continuous function $ f(x)$, periodic with period $ 2 \pi$, such that the Fourier series of $ f(x)$ is divergent at $ x\equal{}0$, but the Fourier series of $ f^2(x)$ is uniformly convergent on $ [0,2 \pi].$ [i]P. Turan[/i]

Show that the equation $x^2 + 8z = 3 + 2y^2$ has no solutions of positive integers $x, y$ and $z$.

Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$. If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac mn$ where $m$ and $n$ are positive coprime integers, find $m+n$.

Prove that equation $y^2=x^3+7$ doesn't have any solution on integers.

Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$. Prove that $m> p$. (Karl Czakler)

Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?

Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$ \[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\] ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)