Is it possible to fill all the cells of the table $9 \times 2002$ with natural numbers so that the sum of the numbers in any column and the sum of the numbers in any string would be prime numbers?

each cells in a $9\times 9 $ grid is painted either blue or red.two cells are called diagonal neighbors if their intersection is exactly a point.show that some cell has exactly two red neighbors,or exactly two blue neighbors, or both.

There are $11$ empty boxes. In one move you can put one coin in some 10 of them. Two people play and take turns. Wins the one after which for the first time there will be $21$ coins in one of the boxes. Who wins when played correctly?

Given a triangle $ABC$ with pairwise distinct sides. on his on the sides, regular triangles $ABC_1$, $BCA_1$, $CAB_1$. are constructed externally. Prove that triangle $A_1B_1C_1$ cannot be regular.

The four-digit number written on the board can be replaced by another, adding one to its two adjacent digits, if neither of these digits is not equal to $9$; or, subtracting one from the adjacent two digits, if none of them is equal to $0$. Is it possible using such operations from does the number $1234$ get the number $2002$?

Each side of the convex quadrilateral was continued into both sides and on all eight extensions set aside equal segments. It turned out that the resulting $8$ points are the outer ends of the construction the given segments are different and lie on the same circle. Prove that the original quadrilateral is a square.

''Moskvich'' and ''Zaporozhets'' drove past the observer on the highway and the Niva moving towards them. It is known that when the Moskvich caught up with the observer, it was equidistant from the Zaporozhets and the Niva, and when the Niva caught up with the observer, it was equal. but removed from ''Moskvich'' and ''Zaporozhets''. Prove that ''Zaporozhets'' at the moment of passing by the observer was equidistant from the Niva and ''Moskvich''.

Among $18$ parts placed in a row, some three in a row weigh $99 $ g each, and all the rest weigh $100$ g each. On a scale with an arrow, identify all $99$-gram parts.

A monic quadratic polynomial $f$ with integer coefficients attains prime values at three consecutive integer points.show that it attains a prime value at some other integer point as well.

In an isosceles triangle $ABC$ ($AB = BC$), point $O$ is the center of the circumcircle. Point $M$ lies on the segment $BO$, point $M' $ is symmetric to $M$ wrt the midpoint of $AB$. Point K is the intersection point of of $M'O$ and $AB$. Point $L$ lies on side BC such that $\angle CLO = \angle BLM$. Prove that points $O, K,B,L$ lie on the same circle

Located on the plane $\left[ \frac43 n \right]$ rectangles with sides parallel to the coordinate axes. It is known that any rectangle intersects at least n rectangles. Prove that exists a rectangle that intersects all rectangles.

Is it possible to arrange the numbers $1, 2, . . . , 60$ in that order, so that the sum of any two numbers between which there is one number, divisible by $2$, the sum of any two numbers between which there are two numbers divisible by $3$, . . . , the sum of any two numbers between which there is are there six numbers, divisible by $7$?

Let $A'$ be a point on one of the sides of the trapezoid $ABCD$ such that line $AA'$ divides the area of the trapezoid in half. Points $B'$, $C'$, $D'$ are defined similarly. Prove that the intersection points of the diagonals of quadrilaterals $ABCD$ and $A'B'C'D'$ are symmetrical wrt the midpoint of midline of trapezoid $ABCD$.

(9.7) On the segment $[0, 2002]$ its ends and the point with coordinate $d$ are marked, where $d$ is a coprime number to $1001$. It is allowed to mark the midpoint of any segment with ends at the marked points, if its coordinate is integer. Is it possible, by repeating this operation several times, to mark all the integer points on a segment? (10.7) On the segment $[0, 2002]$ its ends and $n-1 > 0$ integer points are marked so that the lengths of the segments into which the segment $ [0, 2002]$ is divided are corpime in the total (i.e., have no common divisor greater than $1$). It is allowed to divide any segment with marked ends into $n$ equal parts and mark the division points if they are all integers. (The point can be marked a second time, but it remains marked.) Is it possible, by repeating this operation several times, mark all the integer points on the segment? (11.8) On the segment $ [0,N]$ its ends and $2 $ more points are marked so that the lengths segments into which the segment $[0,N]$ is divided are integer and coprime in total. If there are two marked points $A$ and $B$ such that the distance between them is a multiple of $3$, then we can divide from cutting $AB$ by $3$ equal parts, mark one of the division points and erase one of the points $A, B$. Is it true that for several such actions you can mark any predetermined integer point of the segment $[0,N]$?

What is the largest possible length of an arithmetic progression of positive integers $ a_{1}, a_{2},\cdots , a_{n}$ with difference $ 2$, such that $ {a_{k}}^{2}+1$ is prime for $ k = 1, 2, . . . , n$?

A convex polygon on a plane contains at least $m^2+1$ points with integer coordinates. Prove that it contains $m+1$ points with integer coordinates that lie on the same line.

The perpendicular bisector to side $AC$ of triangle $ABC$ intersects side $BC$ at point $M$ (see fig.). The bisector of angle $\angle AMB$ intersects the circumcircle of triangle $ABC$ at point $K$. Prove that the line passing through the centers of the inscribed circles triangles $AKM$ and $BKM$, perpendicular to the bisector of angle $\angle AKB$.

(10.4) A set of numbers $a_0, a_1,..., a_n$ satisfies the conditions: $a_0 = 0$, $0 \le a_{k+1}- a_k \le 1$ for $k = 0, 1, .. , n -1$. Prove the inequality $$\sum_{k=1}^n a^3_k \le \left(\sum_{k=1}^n a_k \right)^2$$ (11.3) A set of numbers $a_0, a_1,..., a_n$ satisfies the conditions: $a_0 = 0$, $a_{k+1} \ge a_k + 1$ for $k = 0, 1, .. , n -1$. Prove the inequality $$\sum_{k=1}^n a^3_k \ge \left(\sum_{k=1}^n a_k \right)^2$$

Various points $x_1,..., x_n$ ($n \ge 3$) are randomly located on the $Ox$ axis. Construct all parabolas defined by the monic square trinomials and intersecting the Ox axis at these points (and not intersecting axis at other points). Let$ y = f_1$, $...$ , $y = f_m$ are functions that define these parabolas. Prove that the parabola $y = f_1 +...+ f_m$ intersects the $Ox$ axis at two points.

what maximal number of colors can we use in order to color squares of 10 *10 square board so that each row or column contains squares of at most 5 different colors?

The real numbers $x$ and $y$ are such that for any distinct odd primes $p$ and $q$ the number $x^p + y^q$ is rational. Prove that $x$ and $y$ are rational numbers.

The altitude of a quadrangular pyramid $SABCD$ passes through the intersection point of the diagonals of its base $ABCD$. From the tops of the base perpendiculars $AA_1$, $BB_1$, $CC_1$, $DD_1$ are dropped onto lines $SC$, $SD,$ $SA$ and $SB$ respectively. It turned out that the points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are different and lie on the same sphere. Prove that lines $AA_1$, $ BB_1$, $CC_1$, $DD_1$ pass through one point.

Each cell of the checkered plane is colored in one of $n^2$ colors so that in any square of $n \times n$ cells all colors occur. It is known that in some line all the colors occur. Prove that there exists a column colored in exactly $n$ colors.

Let $P(x)$ be a polynomial of odd degree. Prove that the equation $P(P(x)) = 0$ has at least as many different real roots as the equation $P(x) = 0$ original wordingПусть P(x) — многочлен нечетной степени. Докажите, что уравнение P(P(x)) = 0 имеет не меньше различных действительных корней, чем уравнение P(x) = 0

There are $n > 1$ points on the plane. Two take turns connecting more an unconnected pair of points by a vector of one of two possible directions. If after the next move of a player the sum of all drawn vectors is zero, then the second one wins; if it's another move is impossible, and there was no zero sum, then the first one wins. Who wins when played correctly?

Given a convex quadrilateral $ABCD$.Let $\ell_A,\ell_B,\ell_C,\ell_D$ be exterior angle bisectors of quadrilateral $ABCD$. Let $\ell_A \cap \ell_B=K,\ell_B \cap \ell_C=L,\ell_C \cap \ell_D=M,\ell_D \cap \ell_A=N$.Prove that if circumcircles of triangles $ABK$ and $CDM$ be externally tangent to each other then circumcircles of the triangles $BCL$ and $DAN$ are externally tangent to each other.(L.Emelyanov)