Find some nine-digit number $N$, consisting of different digits, such that among all the numbers obtained from $N$ by crossing out seven digits, there would be no more than one prime. Prove that the number found is correct. (If the number obtained by crossing out the digits starts at zero, then the zero is crossed out.)

Two people play this game. At the beginning there are numbers 1, 2, 3, 4 in a circle. With each move, the first one adds 1 to two adjacent numbers, and the second swaps any two adjacent numbers. The first one wins if all numbers become equal. Can the second one interfere with him?

Four drivers took part in the round-robin racing. Their cars started simultaneously from one point and moved at constant speeds. It is known that after the start of the race, for any three cars there was a moment when they met. Prove that after the start of the race there will be a moment when all 4 cars meet. (We consider races to be infinitely long in time.)

Each detail of the “Young Solderer” instructor is a bracket in the shape of the letter $\Pi$, consisting of three single segments. Is it possible from the parts of this constructor are soldered together, a complete wire frame of the cube $2 \times 2 \times 2$, divided into $1 \times 1 \times 1$ cubes? (The frame consists of 27 points, connected by single segments; any two adjacent points must be connected by exactly one piece of wire.) Click to reveal hidden text=original wording]Каждая деталько нструктора ''Юный паяльщик'' — это скобка в виде буквы П, остоящая из трех единичных отрезков. Можно ли издеталей этого конструктора спаятьполный роволочный каркас куба 2 × × 2 × 2, разбитого на кубики 1 × 1 × 1? (Каркас состоит из 27 точек,соединенных единичными отрезками; любые две соседние точки должны бытьсоединены ровно одним проволочным отрезком.)

The product $a_1 \cdot a_2 \cdot ... \cdot a_{100}$ is written on the board , where $a_1$, $a_2$, $ ... $, $a_{100}$, are natural numbers. Let's consider $99$ expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that the values of exactly $32$ of these expressions are even. What is the largest number of even numbers among $a_1$, $a_2$, $ ... $, $a_{100}$ could it be?

In a checkered square $101 \times 101$, each cell of the inner square $99 \times 99$ is painted in one of ten colors (cells adjacent to the border of the square, not painted). Could it turn out that in every in a $3\times 3$ square, is exactly one more cell painted the same color as the central cell?

Segment equal to median $AA_0$ of triangle $ABC$ is drawn from point $A_0$ perpendicular to side $BC$ to the outer side of the triangle. Let's denote the second end of the constructed segment as $A_1$. Points $B_1$ and $C_1$ are constructed similarly. Find the angles of triangle $A_1B_1C_1$ if the angles of the triangle $ABC$ are $30^o$, $30^o$ and $120^o$. original wordingМедиану AA0 треугольника ABC отложили от точки A0 перпендикулярно стороне BC во внешнюю сторону треугольника. Обозначим второй конец построенного отрезка через A1. Аналогично строятся точки B1 и C1. Найдите углы треугольника A1B1C1, если углы треугольника ABC равны 30^o, 30^o и 120^o.

When making a batch of $N \ge 5$ coins, a worker mistakenly made two coins from a different material (all coins look the same). The boss knows that there are exactly two such coins, that they weigh the same, but differ in weight from the others. The employee knows what coins these are and that they are lighter than others. He needs, after carrying out two weighings on cup scales without weights, to convince his boss that the coins are counterfeit easier than real ones, and in which coins are counterfeit. Can he do it?

Each cell of the infinite checkered plane contains one from the numbers $1, 2, 3, 4$ so that each number appears at least once. Let's call a cell correct if the number of distinct numbers written in four adjacent (side) cells to it, equal to the number written in this cell. Can all the cells of the plane be correct?

It is known that $x^2_1+ x^2_2+...+ x^2_6= 6$ and $x_1 + x_2 +....+ x_6 = 0.$ Prove that $ x_1x_2....x_6 \le \frac12$ . .

The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect the circumcircle of this triangle at points $A_0$ and $C_0$, respectively. A straight line passing through the center of the inscribed circle of a triangle $ABC$ is parallel to side $AC$ and intersects line $A_0C_0$ at point $P$. Prove that line $PB$ is tangent to the circumcircle of the triangle $ABC$.

In an acute triangle $ABC$, the angle bisector$AD$ and altitude $BE$ are drawn. Prove that angle $CED$ is greater than $45^o$.

A number $N$ that is not divisible by $81$ can be represented as a sum of squares of three integers divisible by $3$. Prove that it is also representable as the sum of the squares of three integers not divisible by $3$.

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides some triangle.

We call a coloring of an $8\times 8$ board in three colors good if in any corner of five cells contains cells of all three colors. (A five-square corner is a shape made from a $3 \times 3$ square by cutting square $ 2\times 2$.) Prove that the number of good colorings is not less than than $68$.

Given $n > 1$ monic square trinomials $x^2 - a_1x + b_1$,$...$, $x^2-a_nx + b_n$, and all $2n$ numbers are $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n$ are different. Can it happen that each of the numbers $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n is the root of one of these trinomials?

Prove that for every $x$ such that $\sin x \ne 0$, there is such natural $n$, which $$ | \sin nx| \ge \frac{\sqrt3}{2}.$$

Through the point of intersection of the altitudes of an acute triangle $ABC$ three circles pass through, each of which touches one of the sides triangle at the foot of the altitude . Prove that the second intersection points of the circles are the vertices of a triangle similar to the original one.

For what positive integers $n$ are there positive rational, but not integer, numbers $a$ and $b$ such that both numbers $a + b$ and $a^n + b^n$ are integers?

A convex polyhedron has $2n$ faces ($n\ge 3$), and all faces are triangles. What is the largest number of vertices at which converges exactly $3$ edges at a such a polyhedron ?

Product of square trinomials $x^2 - a_1x + b_1$, $x^2 - a_2x + b_2$, $...$, $x^2-a_nx + b_n$ is equal to the polynomial $P(x) = x^{2n} +c_1x^{2n-1} +c_2x^{2n-2} +...+ c_{2n-1}x + c_{2n}$, where the coefficients are $c_1$, $c_2$, $...$ , $c_{2n}$ are positive. Show that for some $k$ ($1\le k \le n$) the coefficients $a_k$ and $b_k$ are positive.

A racing tournament has $12$ stages and $n$ participants. After each stage, all participants, depending on their place $k$, receive points $a_k$ (numbers $a_k$ are natural numbers and $a_1 > a_2 >... > a_n$). At what smallest $n$ can the organizer of the tournament choose numbers $a_1$, $...$ , $a_n$ so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place?

The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect its sides at points $A_1$ and $C_1$, and the circumcircle of this triangle is at points $A_0$ and $C_0$, respectively. Lines $A_1C_1$ and $A_0C_0$ intersect at point P. Prove that the segment connecting $P$ to the center of the incircle of triangle $ABC$ is parallel to $AC$.

In the tetrahedron $ABCD$, perpendiculars $AB'$, $AC'$, $AD'$ are dropped from vertex $A$, on the plane dividing the dihedral angles at the edges $CD$, $BD$, $BC$ in half. Prove that the plane $(B'C'D' )$ is parallel to the plane $(BCD)$.

Prove that if a natural number $N$ is represented in the form as the sum of three squares of integers divisible by $3$, then it is also represented as the sum of three squares of integers not divisible by $3$.

What is the minimum number of cells that can be painted black in white square $300 \times 300$ so that no three black cells form a corner, and after painting any white cell this condition was it violated?