RU-SPB 2021

There are $2021$ points on a circle. Kostya marks a point, then marks the adjacent point to the right, then he marks the point two to its right, then three to the next point's right, and so on. Which move will be the first time a point is marked twice?

K. Kokhas

Misha has a $100$ x $100$ chessboard and a bag with $199$ rooks. In one move he can either put one rook from the bag on the lower left cell of the grid, or remove two rooks which are on the same cell, put one of them on the adjacent square which is above it or right to it, and put the other in the bag. Misha wants to place exactly $100$ rooks on the board, which don't beat each other. Will he be able to achieve such arrangement?

Given is cyclic quadrilateral $ABCD$ with∠ $A = 3$ ∠ $B$ . On the $AB$ side is chosen point $C_1$ , and on side $BC$ - point $A_1$ so that $AA_1 = AC = CC_1$ . Prove that $3A_1C_1>BD$ .

Given are $n$ points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with leading coefficient equal to $1$ . A parabola is called $good$ if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it). What is the greatest number of $good$ parabolas?

Given is an isosceles trapezoid $ABCD$ , such that $AD$ and $BC$ are bases and $AD=2AB$ , and it is inscribed in a circle $c$ . Points $E$ and $F$ are selected on a circle $c$ so that $AC$ || $DE$ and $BD$ || $AF$ . The line $BE$ intersects lines $AC$ and $AF$ at points $X$ and $Y$ , respectively. Prove that the circumcircles of triangles $BCX$ and $EFY$ are tangent to each other.

A school has $450$ students. Each student has at least $100$ friends among the others and among any $200$ students, there are always two that are friends. Prove that $302$ students can be sent on a kayak trip such that each of the $151$ two seater kayaks contain people who are friends.

D. Karpov

For a positive integer $n$ , prove that $$\sum_{n \le p \le n^4} \frac{1}{p} < 4$$ where the sum is taken across primes $p$ in the range $[n, n^4]$ N. Filonov

Solve the following system of equations $$\sin^2{x} + \cos^2{y} = y^4.$$ $$\sin^2{y} + \cos^2{x} = x^2.$$ A. Khrabov

Given are $2021$ prime numbers written in a row. Each number, except for those in the two ends, differs from its two adjacent numbers with $6$ and $12$ . Prove that there are at least two equal numbers.

Given a convex pentagon $ABCDE$ , points $A_1, B_1, C_1, D_1, E_1$ are such that $$AA_1 \perp BE, BB_1 \perp AC, CC_1 \perp BD, DD_1 \perp CE, EE_1 \perp DA.$$ In addition, $AE_1 = AB_1, BC_1 = BA_1, CB_1 = CD_1$ and $DC_1 = DE_1$ . Prove that $ED_1 = EA_1$

Stierlitz wants to send an encryption to the Center, which is a code containing $100$ characters, each a "dot" or a "dash". The instruction he received from the Center the day before about conspiracy reads:

i) when transmitting encryption over the radio, exactly $49$ characters should be replaced with their opposites;

ii) the location of the "wrong" characters is decided by the transmitting side and the Center is not informed of it.

Prove that Stierlitz can send $10$ encryptions, each time choosing some $49$ characters to flip, such that when the Center receives these $10$ ciphers, it may unambiguously restore the original code.

The vertices of a convex $2550$ -gon are colored black and white as follows: black, white, two black, two white, three black, three white, ..., 50 black, 50 white. Dania divides the polygon into quadrilaterals with diagonals that have no common points. Prove that there exists a quadrilateral among these, in which two adjacent vertices are black and the other two are white.

D. Rudenko

A line $\ell$ passes through vertex $C$ of the rhombus $ABCD$ and meets the extensions of $AB, AD$ at points $X,Y$ . Lines $DX, BY$ meet $(AXY)$ for the second time at $P,Q$ . Prove that the circumcircle of $\triangle PCQ$ is tangent to $\ell$ A. Kuznetsov

Kolya found several pairwise relatively prime integers, each of which is less than the square of any other. Prove that the sum of reciprocals of these numbers is less than $2$ .

Let $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Find all $p$ such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum.

A. Khrabov

The cells of a $100 \times 100$ table are colored white. In one move, it is allowed to select some $99$ cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?

S. Berlov

In the pyramid $SA_1A_2 \cdots A_n$ , all sides are equal. Let point $X_i$ be the midpoint of arc $A_iA_{i+1}$ in the circumcircle of $\triangle SA_iA_{i+1}$ for $1 \le i \le n$ with indices taken mod $n$ . Prove that the circumcircles of $X_1A_2X_2, X_2A_3X_3, \cdots, X_nA_1X_1$ have a common point.

The following functions are written on the board, $$F(x) = x^2 + \frac{12}{x^2}, G(x) = \sin(\pi x^2), H(x) = 1.$$ If functions $f,g$ are currently on the board, we may write on the board the functions $$f(x) + g(x), f(x) - g(x), f(x)g(x), cf(x)$$ (the last for any real number $c$ ). Can a function $h(x)$ appear on the board such that $$|h(x) - x| < \frac{1}{3}$$ for all $x \in [1,10]$ ?

A natural number $n$ is given. Prove that $$\sum_{n \le p \le n^2} \frac{1}{p} < 2$$ where the sum is across all primes $p$ in the range $[n, n^2]$

Point $M$ is the midpoint of base $AD$ of an isosceles trapezoid $ABCD$ with circumcircle $\omega$ . The angle bisector of $ABD$ intersects $\omega$ at $K$ . Line $CM$ meets $\omega$ again at $N$ . From point $B$ , tangents $BP, BQ$ are drawn to $(KMN)$ . Prove that $BK, MN, PQ$ are concurrent.

A. Kuznetsov

A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.