For an integer $ n \geq 2 $, let $ s (n) $ be the sum of positive integers not exceeding $ n $ and not relatively prime to $ n $. a) Prove that $ s (n) = \dfrac {n} {2} \left (n + 1- \varphi (n) \right) $, where $ \varphi (n) $ is the number of integers positive cannot exceed $ n $ and are relatively prime to $ n $. b) Prove that there is no integer $ n \geq 2 $ such that $ s (n) = s (n + 2021) $

Two are playing the game "cats and rats" on the chess-board $8\times 8$. The first has one piece -- a rat, the second -- several pieces -- cats. All the pieces have four available moves -- up, down, left, right -- to the neighbour field, but the rat can also escape from the board if it is on the boarder of the chess-board. If they appear on the same field -- the rat is eaten. The players move in turn, but the second can move all the cats in independent directions. a) Let there be two cats. The rat is on the interior field. Is it possible to put the cats on such a fields on the border that they will be able to catch the rat? b) Let there be three cats, but the rat moves twice during the first turn. Prove that the rat can escape.

Show that all terms of the sequence $1,11,111,1111,...$ in base $9$ are triangular numbers, i.e. of the form $\frac{m(m+1)}{2} $for an integer $m$

Given a triangle $ABC$, inside which the point $M$ is marked. On the sides $BC,CA$ and $AB$ the following points $A_1,B_1$ and $C_1$ are chosen, respectively, that $MA_1 \parallel CA$, $MB_1 \parallel AB$, $MC_1 \parallel BC$. Let S be the area of ​​triangle $ABC, Q_M$ be the area of ​​the triangle $A_1 B_1 C_1$. a) Prove that if the triangle $ABC$ is acute, and M is the point of intersection of its altitudes , then $3Q_M \le S$. Is there such a number $k> 0$ that for any acute-angled triangle $ABC$ and the point $M$ of intersection of its altitudes, such thatthe inequality $Q_M> k S$ holds? b) For cases where the point $M$ is the point of intersection of the medians, the center of the inscribed circle, the center of the circumcircle, find the largest $k_1> 0$ and the smallest $k_2> 0$ such that for an arbitrary triangle $ABC$, holds the inequality $k_1S \le Q_M\le k_2S$ (for the center of the circumscribed circle, only acute-angled triangles $ABC$ are considered).

Bokos tribus have $2021$ closed chests, we know that every chest have some amount of rupias and some amount of diamonts. They are going to do a deal with Link, that consits that Link will stay with a amount of chests and Bokos with the rest. Before opening the chests, Link has to say the amount of chest that he will stay with. After this the chests open and Link has to choose the chests with the amount that he previously said. Link doesn´t want to make Bokos angry so he wants to say the smallest number of chest that he will stay with, but guaranteeing that he stay with at least with the half of diamonts, and at least the half of the rupias. What number does Link needs to say?

Find all primes $p$ and $q$, with $p \le q$, so that $$p (2q + 1) + q (2p + 1) = 2 (p^2 + q^2).$$

Find all functions $f : \{1, 2, ... , n,...\} \to Z$ with the following properties (i) if $a, b$ are positive integers and $a | b$, then $f(a) \ge f(b)$; (ii) if $a, b$ are positive integers then $f(ab) + f(a^2 + b^2) = f(a) + f(b)$.

Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.

Let $ l_0,c,\alpha,g$ be positive constants, and let $ x(t)$ be the solution of the differential equation \[ ([l_0\plus{}ct^{\alpha}] ^2x')'\plus{}g[l_0\plus{}ct^{\alpha}] \sin x\equal{}0, \;t \geq 0,\ \;\minus{}\frac{\pi}{2} <x< \frac{\pi}{2},\] satisfying the initial conditions $ x(t_0)\equal{}x_0, \;x'(t_0)\equal{}0$. (This is the equation of the mathematical pendulum whose length changes according to the law $ l\equal{}l_0\plus{}ct^{\alpha}$.) Prove that $ x(t)$ is defined on the interval $ [t_0,\infty)$; furthermore, if $ \alpha >2$ then for every $ x_0 \not\equal{} 0$ there exists a $ t_0$ such that \[ \liminf_{t \rightarrow \infty} |x(t)| >0.\] [i]L. Hatvani[/i]

Is there a described $n$-gon in which each side is longer than the diameter of the inscribed circle a) at $n = 4$? b) when $n = 7$? c) when $n = 6$? [i]P.A.Kozhevnikov[/i]

In a $10\times 10$ square board, half of the squares are coloured white and half black. One side common to two squares on the board side is called a [i]border[/i] if the two squares have different colours. Determine the minimum and maximum possible number of borders that can be on the board.

Prove that if a person a has infinitely many descendants (children, their children, etc.), then a has an infinite sequence $a_0, a_1, \ldots$ of descendants (i.e., $a = a_0$ and for all $n \geq 1, a_{n+1}$ is always a child of $a_n$). It is assumed that no-one can have infinitely many children. [i]Variant 1[/i]. Prove that if $a$ has infinitely many ancestors, then $a$ has an infinite descending sequence of ancestors (i.e., $a_0, a_1, \ldots$ where $a = a_0$ and $a_n$ is always a child of $a_{n+1}$). [i]Variant 2.[/i] Prove that if someone has infinitely many ancestors, then all people cannot descend from $A(dam)$ and $E(ve)$.

Let $n$ be a positive integer divisible by $4$. We consider the permutations $(a_1, a_2,...,a_n)$ of $(1,2,..., n)$ having the following property: for each j we have $a_i + j = n + 1$ where $i = a_j$ . Prove that there are exactly $\frac{ (\frac12 n)!}{(\frac14 n)!}$ such permutations.

Let $ABCD$ be cyclic quadrilateral,let $\gamma$ be it's circumscribed circle and let $M$ be the midpoint of arc $AB$ of $\gamma$,which does not contain points $C,D$.The line that passes through $M$ and the intersection point of diagonals $AC,BD$,intersects $\gamma$ in $N\neq M$. Let $P,Q$ be two points situated on $CD$,such that $\angle{AQD}=\angle{DAP}$ and $\angle{BPC}=\angle{CBQ}$.Prove that circles $\odot(NPQ)$ and $\gamma$ are tangent.

Let $x_1,x_2,...,x_n$ be real numbers in an interval $[m,M]$ such that $\sum_{i=1}^n x_i = 0$. Show that $\sum_{i=1}^n x_i ^2 \le -nmM$

Five members of the Lexington Math Team are sitting around a table. Each flips a fair coin. Given that the probability that three consecutive members flip heads is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Alex Li[/i]

Several blue and green rectangular napkins (perhaps of different sizes) with vertical and horizontal sides were placed on the plane. It turned out that any two napkins of different colors can be crossed by a vertical or horizontal line (perhaps along the border). Prove that it is possible to choose a color, two horizontal lines and one vertical line, so that each napkin of the chosen color is intersected by at least one of the chosen lines.

Find the number of $12$-digit "words" that can be formed from the alphabet $\{0,1,2,3,4,5,6\}$ if neighboring digits must differ by exactly $2$.

Adel draws an $m \times n$ grid of dots on the coordinate plane, at the points of integer coordinates $(a,b)$ where $1 \le a \le m$ and $1 \le b \le n$. He proceeds to draw a closed path along $k$ of these dots, $(a_1, b_1)$,$(a_2,b_2)$,...,$(a_k,b_k)$, such that $(a_i,b_i)$ and $(a_{i+1}, b_{i+1})$ (where $(a_{k+1}, b_{k+1}) = (a_1, b_1)$) are $1$ unit apart for each $1 \le i \le k$. Adel makes sure his path does not cross itself, that is, the $k$ dots are distinct. Find, with proof, the maximum possible value of $k$ in terms of $m$ and $n$.

For each positive integer $n$ let \[x_n=p_1+\cdots+p_n\] where $p_1,\ldots,p_n$ are the first $n$ primes. Prove that for each positive integer $n$, there is an integer $k_n$ such that $x_n<k_n^2<x_{n+1}$

Let $a$ and $b$ be real numbers such that $a + b \ne 0$. Solve the equation $$\frac{1}{(x + a)^2 - b^2} +\frac{1}{(x +b)^2 - a^2}=\frac{1}{x^2 -(a + b)^2}+\frac{1}{x^2-(a -b)^2}$$

Find all primes $p$ such that $p^2-p+1$ is a perfect cube.

A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he draws. How many fish can Bob guarantee getting, no matter Alice's goodwill?

If the number $ 15!$, that is, $ 15 \cdot 14 \cdot 13 \dots 1$, ends with $ k$ zeros when given to the base $ 12$ and ends with $ h$ zeros when given to the base $ 10$, then $ k \plus{} h$ equals: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Let $0<a<1$. $x_1=a,x_2=a^{x_1},\cdots,x_n=a^{x_{n-1}}$. Then sequence $(x_n)$ $\text{(A)}$ Is an increasing sequence. $\text{(B)}$ Is an decreasing sequence. $\text{(C)}$ Increases when $n$ is odd, decreases when $n$ is even. $\text{(D)}$ Decreases when $n$ is odd, increases when $n$ is even.