Show that for any rational number $a$ the equation $y =\sqrt{x^2 +a}$ has infinitely many solutions in rational numbers $x$ and $y$.

Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.

Given two circles $\omega_1$ and $\omega_2$, with centers $O_1$ and $O_2$, respectively intesrecting at two points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_1$. The lines $XA$ and $YA$ intersect $\omega_2$ again in $Z$ and $W$, respectively, such that $A$ is between $X,Z$ and $A$ is between $Y,W$. Let $M$ be the midpoint of $O_1O_2$, S be the midpoint of $XA$ and $T$ be the midpoint of $WA$. Prove that $MS = MT$ if, and only if, the points $X, Y, Z$ and $W$ are concyclic.

Let $n$ be a positive integer. Find the largest integer $N$ for which there exists a set of $n$ weights such that it is possible to determine the mass of all bodies with masses of $1, 2, ..., N$ using a balance scale . (i.e. to determine whether a body with unknown mass has a mass $1, 2, ..., N$, and which namely).

Let $ABCD$ be a parallelogram and let $P{}$ be a point inside it such that $\angle PDA= \angle PBA$. Let $\omega_1$ be the excircle of $PAB$ opposite to the vertex $A{}$. Let $\omega_2$ be the incircle of the triangle $PCD$. Prove that one of the common tangents of $\omega_1$ and $\omega_2$ is parallel to $AD$. [i]Ivan Frolov[/i]

On the square, $1,995$ soldiers lined up in a column, and some of them stood correctly, and some turned backwards. Sergeant Smith remembers only the command "as". With this command, each soldier who sees an even number of faces facing him turns $180^o$, while the rest remain stationary. All movements on command are performed simultaneously. Prove that the sergeant can orient all the soldiers in one direction.

Suppose that point $D$ lies on side $BC$ of triangle $ABC$ such that $AD$ bisects $\angle BAC,$ and let $\ell$ denote the line through $A$ perpendicular to $AD.$ If the distances from $B$ and $C$ to $\ell$ are $5$ and $6,$ respectively, compute $AD.$

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]

Let $ABC$ be a triangle with $BC=30$, $AC=50$, and $AB=60$. Circle $\omega_B$ is the circle passing through $A$ and $B$ tangent to $BC$ at $B$; $\omega_C$ is defined similarly. Suppose the tangent to $\odot(ABC)$ at $A$ intersects $\omega_B$ and $\omega_C$ for the second time at $X$ and $Y$ respectively. Compute $XY$. Let $T = TNYWR$. For some positive integer $k$, a circle is drawn tangent to the coordinate axes such that the lines $x + y = k^2, x + y = (k+1)^2, \dots, x+y = (k+T)^2$ all pass through it. What is the minimum possible value of $k$?

Let $p$ be a constant number such that $0<p<1$. Evaluate \[\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\displaystyle \int_0^1 x^k (1-x)^{2004-k} dx}\]

Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number

Given several squares with the total area $1$. Prove that you can pose them in the square of the area $2$ without any intersections.

Two polygons are cut from the cardboard. Is it possible that for any disposition of these polygons on the plane they have either common inner points or only a finite number of common points on the boundary?

A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Let $x$ and $y$ be nonnegative real numbers such that $x+y=1$. Find the maximum value of $x^4y+xy^4$.

Let $ABC$ be an acute triangle and let $\Gamma$ be its circumcircle. The bisector of $\angle{A}$ intersects $BC$ at $D$, $\Gamma$ at $K$ (different from $A$), and the line through $B$ tangent to $\Gamma$ at $X$. Show that $K$ is the midpoint of $AX$ if and only if $\frac{AD}{DC}=\sqrt{2}$.

The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$. Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums not greater than $9$. [i](I. Gorodnin)[/i]

We have a rectangle with integer sides $m, n$ that is subdivided into $mn$ squares of side $1$. Find the number of little squares that are crossed by the diagonal (without counting those that are touched only in one vertex)

Three cevians are drawn in a triangle that do not intersect at one point. In this case, $4$ triangles and $3$ quadrangles were formed. Find the sum of the areas of the quadrilaterals if the area of each of the four triangles is $8$.

For a positive integer $n$, define $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$. Find the positive integer $k$ for which $7?9?=5?k?$. [i]Proposed by Tristan Shin[/i]

The convex quadrilaterals $ABCD$ and $PQRS$ are made respectively from paper and cardboard. We say that they suit each other if the following two conditions are met : ( 1 ) It is possible to put the cardboard quadrilateral on the paper one so that the vertices of the first lie on the sides of the second, one vertex per side, and (2) If, after this, we can fold the four non-covered triangles of the paper quadrilateral on to the cardboard one, covering it exactly. ( a) Prove that if the quadrilaterals suit each other, then the paper one has either a pair of opposite sides parallel or (a pair of) perpendicular diagonals. (b) Prove that if $ABCD$ is a parallelogram, then one can always make a cardboard quadrilateral to suit it. (N. Vasiliev)

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

Suppose that $A,B,C,D$ are points on a circle, $AB$ is the diameter, $CD$ is perpendicular to $AB$ and meets $AB$ and meets $AB$ at $E , AB$ and $CD$ are integers and $AE - EB=\sqrt{3}$. Find $AE$?

Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$, and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$. Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ $, show that the quadrilateral $ABCD$ is a square.

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic. [i]David Yang.[/i]