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 medians drawn from vertices A and B of triangle ABC are perpendicular. Prove that side AB is the shortest side of ABC.

Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.

Tim has a working analog 12-hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow—at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given instant, the hands on each clock form an angle between $0^\circ$ and $180^\circ$ inclusive. At how many times during that day are the angles on the two clocks equal?

Circle $\omega$ is inscribed in acute triangle $ABC$. Let $I$ denote the center of $\omega$, and let $D,E,F$ be the points of tangency of $\omega$ with $BC, CA, AB$ respectively. Let $M$ be the midpoint of $BC$, and $P$ be the intersection of the line through $I$ perpendicular to $AM$ and line $EF$. Suppose that $AP=9$, $EC=2EA$, and $BD=3$. Find the sum of all possible perimeters of $\triangle ABC$. [i]Proposed by Harry Kim[/i]

[b]5.[/b] Show that a ring $R$ is commutative if for every $x \in R$ the element $x^{2}-x$ belongs to the centre of $R$. [b](A. 18)[/b]

The integers $1, 2,. . . , n$ are arranged in order so that each value is strictly larger than all values above or is strictly less than all values previous. In how many ways can this be done?

Given two circles with one of the centers of the circle is on the other circle. The two circles intersect at two points $C$ and $D$. The line through $D$ intersects the two circles again at $A$ and $ B$. Let $H$ be the midpoint of the arc $AC$ that does not contain $D$ and the segment $HD$ intersects circle that does not contain $H$ at point $E$. Show that $E$ is the center of the incircle of the triangle $ACD$.

Prove that for every positive integer $n$, the number $$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$ is a multiple of $8$.

If $ a_1,...,a_{1988}$ are positive numbers whose arithmetic mean is $ 1988$, show that: $ \sqrt[1988]{\displaystyle\prod_{i,j\equal{}1}^{1988} \left( 1\plus{}\frac{a_i}{a_j} \right)} \ge 2^{1988}$ and determine when equality holds.

An ant is walking on the edges of an icosahedron of side length $1$. Compute the length of the longest path that the ant can take if it never travels over the same edge twice, but is allowed to revisit vertices. [center]<see attached>[/center]

In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game. If after ending of tournament participant have at least $ 75 % $ of maximum possible points he called $winner$ $of$ $tournament$. Find maximum possible numbers of $winners$ $of$ $tournament$.

Let $ N\equal{}100^2\plus{}99^2\minus{}98^2\minus{}97^2\plus{}96^2\plus{}\cdots\plus{}4^2\plus{}3^2\minus{}2^2\minus{}1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $ N$ is divided by $ 1000$.

Find all the polynomials $P(t)$ of one variable that fullfill the following for all real numbers $x$ and $y$: $P(x^2-y^2) = P(x+y)P(x-y)$.

How many ways are there to cover a $3\times 8$ rectangle with $12$ identical dominoes?

The following diagram shows a square where each side has seven dots that divide the side into six equal segments. All the line segments that connect these dots that form a $45^{\circ}$ angle with a side of the square are drawn as shown. The area of the shaded region is 75. Find the area of the original square. [center][img]https://i.snag.gy/Jzx9Fn.jpg[/img][/center]

In convex quadrilateral $ ABCD$, $ CB,DA$ are external angle bisectors of $ \angle DCA,\angle CDB$, respectively. Points $ E,F$ lie on the rays $ AC,BD$ respectively such that $ CEFD$ is cyclic quadrilateral. Point $ P$ lie in the plane of quadrilateral $ ABCD$ such that $ DA,CB$ are external angle bisectors of $ \angle PDE,\angle PCF$ respectively. $ AD$ intersects $ BC$ at $ Q.$ Prove that $ P$ lies on $ AB$ if and only if $ Q$ lies on segment $ EF$.

An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. [quote]For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1[/quote] The number of partitions of n is given by the partition function $p\left ( n \right )$. So $p\left ( 4 \right ) = 5$ . Determine all the positive integers so that $p\left ( n \right )+p\left ( n+4 \right )=p\left ( n+2 \right )+p\left ( n+3 \right )$.

Let $a, b,c$ be positive integers with no common factor and satisfy the conditions $\frac1a +\frac1b=\frac1c$ Prove that $a + b$ is a square.

A magician and his assistant are performing the following trick. There is a row of $13$ empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistant knows which boxes contain coins. The magician returns and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects four boxes, which are then simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always successfully perform the trick. (Igor Zhizhilkin) [url=https://artofproblemsolving.com/community/c6h1801447p11962869]junior version posted here[/url]

Let $ABC$ be a triangle and $X,Y,Z$ be the tangency points of its inscribed circle with the sides $BC, CA, AB$, respectively. Suppose that $C_1, C_2, C_3$ are circle with chords $YZ, ZX, XY$, respectively, such that $C_1$ and $C_2$ intersect on the line $CZ$ and that $C_1$ and $C_3$ intersect on the line $BY$. Suppose that $C_1$ intersects the chords $XY$ and $ZX$ at $J$ and $M$, respectively; that $C_2$ intersects the chords $YZ$ and $XY$ at $L$ and $I$, respectively; and that $C_3$ intersects the chords $YZ$ and $ZX$ at $K$ and $N$, respectively. Show that $I, J, K, L, M, N$ lie on the same circle.

Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$. The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$. The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$. Prove that line $AT$ passes through either $S_1$ or $S_2$. [i]Proposed by Evan Chen[/i]

Find the sum of the $23$ smallest positive integers that are $4$ more than a multiple of $23$ and whose last two digits are $23.$

Does there exist an infinite number of sets $C$ consisting of $1983$ consecutive natural numbers such that each of the numbers is divisible by some number of the form $a^{1983}$, with $a \in \mathbb N, a \neq 1?$

The given integers are $ a_1, a_2, \ldots , a_{11} $ . Prove that there exists a non-zero sequence $ x_1, x_2, \ldots, x_{11} $ with terms from the set $ \{-1,0,1\} $ such that the number $ x_1a_1 + \ldots x_{11}a_{ 11}$ is divisible by 1989.