The $100 \times 100$ table is filled with numbers from $1$ to $10 \ 000$ as shown in the figure. Is it possible to rearrange some numbers so that there is still one number in each cell, and so that the sum of the numbers does not change in all rectangles of three cells?

Given a sequence $a_n$: \[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots \](one '1', two '2' and so on) and another sequence $b_n$ such that $a_{b_n}=b_{a_n}$ for all positive integers $n$. It is known that $b_k=1$ for some $k>100$. Prove that $b_m=1$ for all $m>k$.

In unequal triangle $ABC$ bisector $AK$ was drawn. Diameter $XY$ of its circumcircle is perpendicular to $AK$ (order of points on circumcircle is $B-X-A-Y-C$). A circle, passing on points $X$ and $Y$, intersect segments $BK$ and $CK$ in points $T$ and $Z$ respectively. Prove that if $KZ=KT$, then $XT \perp YZ$.

Given a $101$-digit number $a$ and an arbitrary positive integer $b$. Prove that there is at most a $102$-digit positive integer $c$ such that any number of the form $\overline{caaa \dots ab}$ is composite.

There are $100$ points of general position marked on the plane (i.e. no three lie on the same straight line). Prove that it is possible to select three marked points $A, B, C$ so that for any point $D$ of the remaining $97$ marked points, the lines $AD$ and $CD$ would not contain points lying inside the triangle $ABC$.

Polynomial $P(x)$ with integer coefficients is given. For some positive integer $n$ numbers $P(0),P(1),\dots,P(2^n+1)$ are all divisible by $2^{2^n}$. Prove that values of $P(x)$ in all integer points are divisible by $2^{2^n}$.

A tourist has arrived on an island where $100$ wizards live, each of whom can be a knight or a liar. He knows that at the time of his arrival, one of the hundred wizards is a knight (but does not know who exactly), and the rest are liars. A tourist can choose any two wizards $A$ and $B$ and ask $A$ to spell on $B$ with the spell "Whoosh"!, which changes the essence (turns a knight into a liar, and a liar into a knight). Wizards fulfill the tourist's requests, but if at that moment wizard $A$ is a knight, then the essence of $B$ really changes, and if $A$ is a liar, that doesn't change. The tourist wants to know the essence of at least $k$ wizards at the same time after several consecutive requests. For which maximum $k$ will he be able to achieve his goal?

In the cells of the $2024\times 2024$ board, integers are arranged so that in any $2 \times 2023$ rectangle (vertical or horizontal) with one cut corner cell that does not go beyond the board, the sum of the numbers is divided by $13$. Prove that the sum of all the numbers on the board is divisible by $13$.

$32$ real and $32$ fake coins are given the same appearance. All fake coins weigh equally and less than the real ones, which also all weigh the same. How to determine the type of at least seven coins in six weighings on a scale with two bowls?

On the side $BC$ of acute triangle $ABC$ point $P$ was chosen. Point $E$ is symmetric to point $B$ onto line $AP$. Segment $PE$ meets circumcircle of triangle $ABP$ in point $D$. $M$ is midpoint of side $AC$. Prove that $DE+AC>2BM$.

Let's consider all possible quadratic trinomials of the form $x^2 + ax + b$, where $a$ and $b$ are positive integers not exceeding some positive integer $N$. Prove that the number of pairs of such trinomials having a common root does not exceed $N^2$.

$2 \ 000 \ 000$ points with integer coordinates are marked on the numeric axis. Segments of lengths $97$, $100$ and $103$ with ends at these points are considered. What is the largest number of such segments?

Inscribed hexagon $AB_1CA_1BC_1$ is given. Circle $\omega$ is inscribed in both triangles $ABC$ and $A_1B_1C_1$ and touches segments $AB$ and $A_1B_1$ at points $D$ and $D_1$ respectively. Prove that if $\angle ACD = \angle BCD_1$, then $\angle A_1C_1D_1 = \angle B_1C_1D$.

The edges of a complete graph on $1000$ vertices are colored in three colors. Prove that this graph contains a non-self-intersecting single-color cycle whose length is odd and not less than $41$.

Dima has red and blue felt—tip pens, with one of them he paints rational points on the numerical axis, and with the other - irrational ones. Dima colored $100$ rational and $100$ irrational points, after which he erased the signatures that allowed to find out where the origin was and what the scale was. Sergey has a compass with which he can measure the distance between any two colored points $A$ and $B$, and then mark on the axis a point located at a measured distance from any colored point $C$ (left or right); at the same time, Dima immediately paints it with the appropriate felt-tip pen. How Sergei can find out what color Dima paints rational points and what color he paints irrational ones?

A strongman Bambula can carry several weights at the same time, if their total weight does not exceed $200$ kg, and these weights are no more than three. On the way to work, he injured his finger and found that he could now carry no more than two weights (and still no more than $200$ kg). At what minimum $k$ is the statement true: any set of $100$ weights that Bambula could previously carry in $50$ runs, with a sore finger, he will be able to carry in no more than $k$ runs?

The triangle $ABC$ is inscribed in a circle. Two ants crawl out of points $B$ and $C$ at the same time. They crawl along the arc $BC$ towards each other so that the product of the distances from them to point $A$ remains unchanged. Prove that during their movement (until the moment of meeting), the straight line passing through the ants touches some fixed circle.

The coach lined up $200$ volleyball players and gave them $m$ balls (each volleyball player could get any number of balls). From time to time, one of the volleyball players throws the ball to another (and he catches it). After a while, it turned out that of any two volleyball players, the left one threw the ball to the right exactly twice, and the right one to the left exactly once. For which minimum $m$ is this possible?

Let $AH$ be altitude in acute trinagle $ABC$, inscribed in circle $s$. Points $D$ and $E$ are chosen on segment $BH$. Points $X$ and $Y$ are chosen on rays $AD$ and $AE$, respectively, such that midpoints of segments $DX$ and $EY$ lies on $s$. Suppose that points $B$, $X$, $Y$ and $C$ are concyclic. Prove that $BD+BE=2CH$.

Call a positive integer number $n$ poor if equation \[x_1x_2 \dots x_{101}=(n-x_1)(n-x_2)\dots (n-x_{101}) \]has no solutions in positive integers $1<x_i<n$. Does there exist poor number, which has more than $100 \ 000$ distinct prime divisors?

In a very large City, they are building a subway: there are many stations, some pairs of which are connected by tunnels, and from any station you can get through tunnels to any other. All metro tunnels must be divided into "lines": each line consists of several consecutive tunnels, all stations in which are different (in particular, the line should not be circular); lines consisting of one tunnel are also allowed. By law, it is required that you can get from any station to any other station by making no more than $100$ transfers from line to line. At what is the largest $N$, any connected metro with $N$ stations can be divided into lines, observing the law?