Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}9\equal{}\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.$

Let $ABC$ be an acute-angled triangle, with $AC \neq BC$ and let $O$ be its circumcenter. Let $P$ and $Q$ be points such that $BOAP$ and $COPQ$ are parallelograms. Show that $Q$ is the orthocenter of $ABC$.

Let $I$ be the incenter of the non-isosceles triangle $ABC$ and let $A',B',C'$ be the tangency points of the incircle with the sides $BC,CA,AB$ respectively. The lines $AA'$ and $BB'$ intersect in $P$, the lines $AC$ and $A'C'$ in $M$ and the lines $B'C'$ and $BC$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular. [i]Alternative formulation.[/i] The incircle of a non-isosceles triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$ and $AB$ in $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, respectively. The lines $AA^{\prime}$ and $BB^{\prime}$ intersect in $P$, the lines $AC$ and $A^{\prime}C^{\prime}$ intersect in $M$, and the lines $BC$ and $B^{\prime}C^{\prime}$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular.

Let $P$ be a fixed point and $T$ a given triangle that contains the point $P$. Translate the triangle $T$ by a given vector $\bold{v}$ and denote by $T'$ this new triangle. Let $r, R$, respectively, be the radii of the smallest disks centered at $P$ that contain the triangles $T , T'$, respectively. Prove that $r + |\bold{v}| \leq 3R$ and find an example to show that equality can occur.

Let $a_1, b_1, a_2, b_2, \dots , a_n, b_n$ be nonnegative real numbers. Prove that \[ \sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}. \]

Let $A$ be a $3\times 3$ real matrix such that the vectors $Au$ and $u$ are orthogonal for every column vector $u\in \mathbb{R}^{3}$. Prove that: a) $A^{T}=-A$. b) there exists a vector $v \in \mathbb{R}^{3}$ such that $Au=v\times u$ for every $u\in \mathbb{R}^{3}$, where $v \times u$ denotes the vector product in $\mathbb{R}^{3}$.

In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.

Given some points in the plane. For some pairs $A,B$ the vector $AB$ is chosen. For every point the number of the chosen vectors starting in that point equal to the number of the chosen vectors ending in that point. Prove that the sum of the chosen vectors equals to zero vector.

Zeroes and ones are arranged in all the squares of $n\times n$ table. All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form [asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy] (consisting of a square and its neighbours from left and from below) is even. Prove that no two rows of the table are identical. [i]Proposed by O. Vanyushina[/i]

In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d$ are positive integers and $d$ is not divisible by the square of any prime number. Find $m+n+d.$

An $n\times n$ matrix $A$ with integer entries is called [i]representative[/i] if, for any integer vector $\mathbf{v}$, there is a finite sequence $0=\mathbf{v}_0,\mathbf{v}_1,\dots,\mathbf{v}_{\ell}=\mathbf{v}$ of integer vectors such that for each $0\leq i <\ell$, either $\mathbf{v}_{i+1}=A\mathbf{v}_{i}$ or $\mathbf{v}_{i+1}-\mathbf{v}_i$ is an element of the standard basis (i.e. one of its entries is $1$, the rest are all equal to $0$). Show that $A$ is not representative if and only if $A^T$ has a real eigenvector with all non-negative entries and non-negative eigenvalue.

Let $p$ be a prime and let $M$ be an $n\times m$ matrix with integer entries such that $Mv\not\equiv 0\pmod{p}$ for any column vector $v\neq 0$ whose entries are $0$ are $1$. Show that there exists a row vector $x$ with integer entries such that no entry of $xM$ is $0\pmod{p}$. (translated by L. Erdős)

Find the flux of the vector field $\overrightarrow{r}/r^3$ through the surface \[(x-1)^2+y^2+z^2=2\]

$ \Delta_1,\ldots,\Delta_n$ are $ n$ concurrent segments (their lines concur) in the real plane. Prove that if for every three of them there is a line intersecting these three segments, then there is a line that intersects all of the segments.

Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$. For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

Let $n$ be an integer greater than 2, and $P_1, P_2, \cdots , P_n$ distinct points in the plane. Let $\mathcal S$ denote the union of all segments $P_1P_2, P_2P_3, \dots , P_{n-1}P_{n}$. Determine if it is always possible to find points $A$ and $B$ in $\mathcal S$ such that $P_1P_n \parallel AB$ (segment $AB$ can lie on line $P_1P_n$) and $P_1P_n = kAB$, where (1) $k = 2.5$; (2) $k = 3$.

Let $ a,b,c$ be positive real numbers. Prove the inequality: $ \frac {a^3}{b^2} \plus{} \frac {b^3}{c^2} \plus{} \frac {c^3}{a^2}\ge \frac {a^2}{b} \plus{} \frac {b^2}{c} \plus{} \frac {c^2}{a}$

A river flows between two houses $ A$ and $ B$, the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from $ A$ to $ B$, using the bridge to cross the river may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks.

Decompose a $5$-dimensional real linear space into the irreducible invariant subspaces of the group generated by cyclic permutations of the basis vectors.

On the coordinate plane, at some points with integer coordinates, there is a pebble (a finite number of pebbles). It is allowed to make the following move: select a pair of pebbles, take some vector $\vec{a}$ with integer coordinates and then move one of the selected pebbles to vector $\vec{a}$, and the other to the opposite vector $-\vec{a}$; it is forbidden that there should be more than one pebble at one point. Is it always possible to achieve a situation in which all the pebbles lie on the same straight line in a few moves? [i] K. Ivanov [/i]

On the sides $BC$, $CA$, $AB$ of an acute-angled triangle $ABC$, we erect (outwardly) the squares $BB_aC_aC$, $CC_bA_bA$, $AA_cB_cB$, respectively. On the sides $B_cB_a$ and $C_aC_b$ of the triangles $BB_cB_a$ and $CC_aC_b$, we erect (outwardly) the squares $B_cB_vB_uB_a$ and $C_aC_uC_vC_b$. Prove that $B_uC_u\parallel BC$. [i]Comment.[/i] This problem originates in the 68th Moscow MO 2005, and a solution was posted in http://www.mathlinks.ro/Forum/viewtopic.php?t=30184 . However ingenious this solution is, there is a different one which shows a bit more: $B_uC_u=4\cdot BC$. Darij

Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?

There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.