We call the $main$ $divisors$ of a composite number $n$ the two largest of its natural divisors other than $n$. Composite numbers $a$ and $b$ are such that the main divisors of $a$ and $b$ coincide. Prove that $a=b$.

Given is triangle $ABC$ with incenter $I$ and $A$-excenter $J$. Circle $\omega_b$ centered at point $O_b$ passes through point $B$ and is tangent to line $CI$ at point $I$. Circle $\omega_c$ with center $O_c$ passes through point $C$ and touches line $BI$ at point $I$. Let $O_bO_c$ and $IJ$ intersect at point $K$. Find the ratio $IK/KJ$.

$200$ natural numbers are written in a row. For any two adjacent numbers of the row, the right one is either $9$ times greater than the left one, $2$ times smaller than the left one. Can the sum of all these 200 numbers be equal to $24^{2022}$?

There are $18$ children in the class. Parents decided to give children from this class a cake. To do this, they first learned from each child the area of the piece he wants to get. After that, they showed a square-shaped cake, the area of which is exactly equal to the sum of $18$ named numbers. However, when they saw the cake, the children wanted their pieces to be squares too. The parents cut the cake with lines parallel to the sides of the cake (cuts do not have to start or end on the side of the cake). For what maximum k the parents are guaranteed to cut out $k$ square pieces from the cake, which you can give to $k$ children so that each of them gets what they want?

Given an infinite sequence of numbers $a_1, a_2,...$, in which there are no two equal members. Segment $a_i, a_{i+1}, ..., a_{i+m-1}$ of this sequence is called a monotone segment of length $m$, if $a_i < a_{i+1} <...<a_{i+m-1}$ or $a_i > a_{i+1} >... > a_{i+m-1}$. It turned out that for each natural $k$ the term $a_k$ is contained in some monotonic segment of length $k + 1$. Prove that there exists a natural $N$ such that the sequence $a_N , a_{N+1} ,...$ monotonic.

What is the smallest natural number $a$ for which there are numbers $b$ and $c$ such that the quadratic trinomial $ax^2 + bx + c$ has two different positive roots not exceeding $\frac {1}{1000}$?

There are $998$ cities in a country. Some pairs of cities are connected by two-way flights. According to the law, between any pair cities should be no more than one flight. Another law requires that for any group of cities there will be no more than $5k+10$ flights connecting two cities from this group, where $k$ is the number number of cities in the group. Prove that several new flights can be introduced so that laws still hold and the total number of flights in the country is equal to $5000$.

A circle $\omega$ is inscribed in triangle $ABC$, tangent to the side $BC$ at point $K$. Circle $\omega'$ is symmetrical to the circle $\omega$ with respect to point $A$. The point $A_0$ is chosen so that the segments $BA_0$ and $CA_0$ touch $\omega'$. Let $M$ be the midpoint of side $BC$. Prove that the line $AM$ bisects the segment $KA_0$.

On side $BC$ of an acute triangle $ABC$ are marked points $D$ and $E$ so that $BD = CE$. On the arc $DE$ of the circumscribed circle of triangle $ADE$ that does not contain the point $A$, there are points $P$ and $Q$ such that $AB = PC$ and $AC = BQ$. Prove that $AP=AQ$.

Initially, a pair of numbers $(1,1)$ is written on the board. If for some $x$ and $y$ one of the pairs $(x, y-1)$ and $(x+y, y+1)$ is written on the board, then you can add the other one. Similarly for $(x, xy)$ and $(\frac {1} {x}, y)$. Prove that for each pair that appears on the board, its first number will be positive.

Given is a natural number $n>4$. There are $n$ points marked on the plane, no three of which lie on the same line. Vasily draws one by one all the segments connecting pairs of marked points. At each step, drawing the next segment $S$, Vasily marks it with the smallest natural number, which hasn't appeared on a drawn segment that has a common end with $S$. Find the maximal value of $k$, for which Vasily can act in such a way that he can mark some segment with the number $k$?

There are $11$ integers (not necessarily distinct) written on the board. Can it turn out that the product of any five of them is greater than the product of the other six?

Given is a natural number $n > 5$. On a circular strip of paper is written a sequence of zeros and ones. For each sequence $w$ of $n$ zeros and ones we count the number of ways to cut out a fragment from the strip on which is written $w$. It turned out that the largest number $M$ is achieved for the sequence $11 00...0$ ($n-2$ zeros) and the smallest - for the sequence $00...011$ ($n-2$ zeros). Prove that there is another sequence of $n$ zeros and ones that occurs exactly $M$ times.

Point $E$ is marked on side $BC$ of parallelogram $ABCD$, and on the side $AD$ - point $F$ so that the circumscribed circle of $ABE$ is tangent to line segment $CF$. Prove that the circumcircle of triangle $CDF$ is tangent to line $AE$.

For a natural number $N$, consider all distinct perfect squares that can be obtained from $N$ by deleting one digit from its decimal representation. Prove that the number of such squares is bounded by some value that doesn't depend on $N$.

In the coordinate plane,the graps of functions $y=sin x$ and $y=tan x$ are drawn, along with the coordinate axes. Using compass and ruler, construct a line tangent to the graph of sine at a point above the axis, $Ox$, as well at a point below that axis (the line can also meet the graph at several other points)

An acute-angled triangle $ABC$ is fixed on a plane with largest side $BC$. Let $PQ$ be an arbitrary diameter of its circumscribed circle, and the point $P$ lies on the smaller arc $AB$, and the point $Q$ is on the smaller arc $AC$. Points $X, Y, Z$ are feet of perpendiculars dropped from point $P$ to the line $AB$, from point $Q$ to the line $AC$ and from point $A$ to line $PQ$. Prove that the center of the circumscribed circle of triangle $XYZ$ lies on a fixed circle.

Given is natural number $n$. Sasha claims that for any $n$ rays in space, no two of which have a common point, he will be able to mark on these rays $k$ points lying on one sphere. What is the largest $k$ for which his statement is true?

From each vertex of triangle $ABC$ we draw two rays, red and blue, symmetric about the angle bisector of the corresponding angle. The circumcircles of triangles formed by the intersection of rays of the same color. Prove that if the circumcircle of triangle $ABC$ touches one of these circles then it also touches to the other one.