At 12 o'clock in the afternoon, "Zaporozhets" and "Moskvich" were at a distance of 90 km and began to move towards each other at a constant speed. Two hours later they were again at a distance of 90 km. Dunno claims that ''Zaporozhets'' before meeting with ''Moskvich'' and ''Moskvich'' after the meeting with ''Zaporozhets'' , have drove a total of 60 km. Prove that he is wrong. original wordingВ 12 часов дня ''Запорожец'' и ''Москвич'' находилисьна расстоянии 90 км и начали двигаться навстречу друг другу с постоянной скоростью. Через два часа они снова оказались на расстоянии 90 км. Незнайка утверждает, что ''Запорожец'' до встречи с ''Москвичом'' и ''Москвич'' после встречи с ''Запорожцем'' проехали в сумме 60 км. Докажите, что он не прав.

In the middle cell of the $1 \times 2005$ strip there is a chip. Two players each queues move it: first, the first player moves the piece one cell in any direction, then the second one moves it $2$ cells, the $1$st - by $4$ cells, the 2nd by $8$, etc. (the $k$-th shift occurs by $2^{k-1}$ cells). That, whoever cannot make another move loses. Who can win regardless of the opponent's play?

Given an acute triangle $ABC$. Points $B'$ and $C'$ are symmetrical, respectively, to vertices $B$ and $ C$ wrt straight lines $AC$ and $AB$. Let $P$ be the intersection point of the circumcircles of triangles $ABB'$ and $ACC'$, different from $A$. Prove that the center of the circumcircle of triangle $ABC$ lies on line $PA$.

It is known that the sum of the digits of the natural number $N$ is $100$, and the sum of the digits of the number $5N$ is $50$. Prove that $N$ is even.

In quadrilateral $ABCD$, angles $A$ and $C$ are equal. Angle bisector of $B$ intersects line $AD$ at point $P$. Perpendicular on $BP$ passing through point $A$ intersects line $BC$ at point $Q$. Prove that the lines $PQ$ and $CD$ are parallel.

Find all pairs $(x, y)$ of natural numbers such that $$x + y = a^n, x^2 + y^2 = a^m$$for some natural $a, n, m$.

8.8, 9.8, 11.8 a) 99 boxes contain apples and oranges. Prove that we can choose 50 boxes in such a way that they contain at least half of all apples and half of all oranges. b) 100 boxes contain apples and oranges. Prove that we can choose 34 boxes in such a way that they contain at least a third of all apples and a third of all oranges. c) 100 boxes contain apples, oranges and bananas. Prove that we can choose 51 boxes in such a way that they contain at least half of all apples, and half of all oranges and half of all bananas. (I. Bogdanov, G. Chelnokov, E. Kulikov)

Five teams participated in the commercial football tournament. Each had to play exactly one match with each other. Due to financial difficulties, the organizers canceled some games. In the end It turned out that all teams scored a different number of points and not a single team in the points column had a zero. What is the smallest number of games could be played in a tournament if three points were awarded for a win, for a draw - one, for a defeat - zero?

9.2 Given 19 cards. Is it possible to write a nonzero digit on each card in such a way that you can compose from these cards an unique 19-digits number, which is divisible by 11? (R. Zhenodarov, I. Bogdanov)

Two players take turns placing the numbers $1, 2, 3,. . . , 24$, in each of the $24$ squares on the surface of a $2 \times 2 \times 2$ cube (each number can be placed once). The second player wants the sum of the numbers in each cell the rings of $8$ cells encircling the cube were identical. Will he be able to the first player to stop him?

9.4, 10.3 Let $I$ be an incenter of $ABC$ ($AB<BC$), $M$ is a midpoint of $AC$, $N$ is a midpoint of circumcircle's arc $ABC$. Prove that $\angle IMA=\angle INB$. (A. Badzyan)

9.6, 10.6 Construct for each vertex of the trapezium a symmetric point wrt to the diagonal, which doesn't contain this vertex. Prove that if four new points form a quadrilateral then it is a trapezium. (L. Emel'yanov)

9.7 Is there an infinite arithmetic sequence $\{a_n\}\subset \mathbb N$ s.t. $a_n+...+a_{n+9}\mid a_n...a_{n+9}$ for all $n$? (V. Senderov)

The cosines of the angles of one triangle are respectively equal to the sines of the angles of the other triangle. Find the largest of these six angles of triangles.

10.2 Prove for all $x>0$ and $n\in\mathbb{N}$ the following inequality \[1+x^{n+1}\geq \frac{(2x)^n}{(1+x)^{n-1}}.\] (A. Khrabrov)

10.4, 11.3 Given $N\geq 3$ points enumerated with 1, 2, ..., $N$. Each two numbers are connected by mean of arrow from a lesser number to a greater one. A coloring of all arrows into red and blue is called monochromatic iff for any numbers $A$ and $B$ there are no two monochromatic paths from $A$ to $B$ of different colors. Find the number of monochromatic colorings. (I. Bogdanov, G. Chelnokov)

Arithmetic progression $a_1, a_2, . . . , $ consisting of natural numbers is such that for any $n$ the product $a_n \cdot a_{n+31}$ is divisible by $2005$. Is it possible to say that all terms of the progression are divisible by $2005$?

10.7 Find all pairs $(a,b)$ of natural numbers s.t. $a^n+b^n$ is a perfect $n+1$th power for all $n\in\mathbb{N}$. (V. Senderov)

A rectangle is drawn on checkered paper, the sides of which form angles of $45^o$ with the grid lines, and the vertices do not lie on the grid lines. Can an odd number of grid lines intersect each side of a rectangle?

Find all pairs of numbers $x, y \in \left( 0, \frac{\pi}{2}\right)$ , satisfying the equality $$\sin x + \sin y = \sin (xy)$$

It is known that there is a number $S$ such that if $ a+b+c+d = S$ and $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d} = S$ $(a, b, c, d$ are different from zero and one$)$, then $\frac{1}{a- 1} ++ \frac{1}{b- 1} + \frac{1}{c- 1} + \frac{1}{d -1} = S.$ Find $S$.

11.4 Let $AA_1$ and $BB_1$ are altitudes of an acute non-isosceles triangle $ABC$, $A'$ is a midpoint of $BC$ and $B'$ is a midpoint of $AC$. A segement $A_1B_1$ intersects $A'B'$ at point $C'$. Prove that $CC'\perp HO$, where $H$ is a orthocenter and $O$ is a circumcenter of $ABC$. (L. Emel'yanov)

Prove that for any polynomial $P$ with integer coefficients and any natural number $k$ there exists a natural number $n$ such that $P(1) + P(2) + ...+ P(n)$ is divisible by $k$.

11.6 Construct for each vertex of the quadrilateral of area $S$ a symmetric point wrt to the diagonal, which doesn't contain this vertex. Let $S'$ be an area of the obtained quadrilateral. Prove that $\frac{S'}{S}<3$. (L. Emel'yanov)

11.7 Let $N$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 7, and $M$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 8. Compare $M$ and $N$. (A. Golovanov)