The numbers $1, 2, ..., 2020$ and $2021$ are written on a blackboard. The following operation is executed: Two numbers are chosen, both are erased and replaced by the absolute value of their difference. This operation is repeated until there is only one number left on the blackboard. (a) Show that $2021$ can be the final number on the blackboard. (b) Show that $2020$ cannot be the final number on the blackboard. (Karl Czakler)

Points $A,M,B,C,D$ are given on a circle in this order such that $A$ and $B$ are equidistant from $M$. Lines $MD$ and $AC$ intersect at $E$ and lines $MC$ and $BD$ intersect at $F$. Prove that the quadrilateral $CDEF$ is inscridable in a circle.

Determine the smallest positive integer $n$ whose prime factors are all greater than $18$, and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$.

Is it possible to fill a rectangular table with black and white squares (only) so, that the number of black squares will equal to the number of white squares, and each row and each column will have more than $75\%$ squares of the same colour?

A student will take an exam in which they have to solve three chosen problems by chance of a list of $10$ possible problems. It will be approved if it correctly resolves two problems. Considering that the student can solve five of the problems on the list and not know how to solve others, how likely is he to pass the exam?

Find all pairs $(x, y)$ of rational numbers such that $y^2 =x^3 -3x+2$.

Let $a, b$, and $c$ be positive real numbers. Prove: $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\le \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2}$ .

A sequence of primes $p_1, p_2, \dots$ is given by two initial primes $p_1$ and $p_2$, and $p_{n+2}$ being the greatest prime divisor of $p_n + p_{n+1} + 2018$ for all $n \ge 1$. Prove that the sequence only contains finitely many primes for all possible values of $p_1$ and $p_2$.

Given a positive integer $n$. (a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$, \[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\] (b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.

Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$ and $v=2xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is: [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)--(1,0)--(0,-2)--(-1,0)--cycle); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),SE); label("$(0,-2)$",(0,-2),SE); label("$(-1,0)$",(-1,0),SW); label("$\textbf{(A)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)..(1,0)..(0,-2)^^(0,-2)..(-1,0)..(0,2)); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),SE); label("$(0,-2)$",(0,-2),SE); label("$(-1,0)$",(-1,0),SW); label("$\textbf{(B)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)--(1,0)--(-1,0)--cycle); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),S); label("$(-1,0)$",(-1,0),S); label("$\textbf{(C)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)..(1/2,3/2)..(1,0)--(-1,0)..(-1/2,3/2)..(0,2)); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),S); label("$(-1,0)$",(-1,0),S); label("$\textbf{(D)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,1)--(1,0)--(0,-1)--(-1,0)--cycle); label("$(0,1)$",(0,1),NE); label("$(1,0)$",(1,0),SE); label("$(0,-1)$",(0,-1),SE); label("$(-1,0)$",(-1,0),SW); label("$\textbf{(E)}$",(-2,1.5)); [/asy]

Find the integer $x$ for which $135^3+138^3=x^3-1.$

Let the incircle of triangle $ABC$ touches the sides $AB,BC,CA$ in $C_1,A_1,B_1$ respectively. If $A_1C_1$ cuts the parallel to $BC$ from $A$ at $K$ prove that $\angle KB_1A_1=90.$

Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, "At least two of us have the same score." Marty thinks, "I didn't get the lowest score." Shana thinks, "I didn't get the highest score." List the scores from lowest to highest for Marty (M), Quay (Q) and Shana (S). $ \text{(A)}\ \text{S,Q,M}\qquad\text{(B)}\ \text{Q,M,S}\qquad\text{(C)}\ \text{Q,S,M}\qquad\text{(D)}\ \text{M,S,Q}\qquad\text{(E)}\ \text{S,M,Q} $

A circle and a point $P$ inside it are given. Two arbitrary perpendicular rays starting at point $P$ intersect the circle at points $A$ and $B$. Point $X$ is the projection of point $P$ onto line $AB, Y$ is the intersection point of tangents to the circle drawn through points $A$ and $B$. Prove that all lines $XY$ pass through the same point. (A. Zaslavsky)

Find five positive integers such that the greatest common divisor of each pair is equal to the difference between them. (SI Tokarev)

The first four terms of an infinite sequence $S$ of decimal digits are $1$, $9$, $8$, $2$, and succeeding terms are given by the final digit in the sum of the four immediately preceding terms. Thus $S$ begins $1$, $9$, $8$, $2$, $0$, $9$, $9$, $0$, $8$, $6$, $3$, $7$, $4$, $\cdots$. Do the digits $3$, $0$, $4$, $4$ ever come up consecutively in $S$?

Find all integers $ n\ge 2$ with the following property: There exists three distinct primes $p,q,r$ such that whenever $ a_1,a_2,a_3,\cdots,a_n$ are $ n$ distinct positive integers with the property that at least one of $ p,q,r$ divides $ a_j - a_k \ \forall 1\le j\le k\le n$, one of $ p,q,r$ divides all of these differences.

An integer $n > 2$ is given. Peter wants to draw $n{}$ arcs of length $\alpha{}$ of great circles on a unit sphere so that they do not intersect each other. Prove that [list=a] [*]for all $\alpha<\pi+2\pi/n$ it is possible; [*]for all $\alpha>\pi+2\pi/n$ it is impossible; [/list] [i]Ilya Bogdanov[/i]

Let $ABCD$ be a cyclic quadrilateral with $AB = 1$, $BC = 2$, $CD = 3$, $DA = 4$. Find the square of the area of quadrilateral $ABCD$.

For a positive integer $n$ and a nonzero digit $d$, let $f(n, d)$ be the smallest positive integer $k$, such that $kn$ starts with $d$. What is the maximal value of $f(n, d)$, over all positive integers $n$ and nonzero digits $d$?

In the triangle $ABC$, $\angle A=90^\circ$, the bisector of $\angle B$ meets the altitude $AD$ at the point $E$, and the bisector of $\angle CAD$ meets the side $CD$ at $F$. The line through $F$ perpendicular to $BC$ intersects $AC$ at $G$. Prove that $B,E,G$ are collinear.

A convex quadrilateral $ABCD$ is given. There rays $BA$ and $CD$ meet in $P$, and the rays $BC$ and $AD$ meet in $Q$. Let $H$ be the projection of $D$ on $PQ$. Prove that $ABCD$ is cyclic if and only if the angle between the rays beginning at $H$ and tangent to the incircle of triangle $ADP$ is equal to the angle between the rays beginning at $H$ and tangent to the incircle of triangle $CDQ$. Also find out whether $ABCD$ is inscribable or circumscribable and justify.

In an acute triangle $\triangle ABC, \angle A > \angle B$. Let the midpoint of $AB$ be $D$, and let the foot of the perpendicular from $A$ to $BC$ be $E$, and $B$ from $CA$ be $F$. Let the circumcenter of $\triangle DEF$ be $O$. A point $J$ on segment $BE$ satisfies $\angle ODC = \angle EAJ$. Prove that $AJ \cap DC$ lies on the circumcircle of $\triangle BDE$.

Find all the integer solutions $(x,y,z)$ of the equation $(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)$,

[b]7.1. / 6.5[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers. [b]7.2[/b] Given a circle $O$ and a square $K$, as well as a line $L$. Construct a segment of given length parallel to $L$ and such that its ends lie on $O$ and $K$ respectively [b]7.3[/b] The three-digit number $\overline{abc}$ is divisible by $37$. Prove that the sum of the numbers $\overline{bca}$ and $\overline{cab}$ is also divisible by $37$.[b] (typo corrected)[/b] [b]7.4.[/b] Point $C$ is the midpoint of segment $AB$. On an arbitrary ray drawn from point $C$ and not lying on line $AB$, three consecutive points $P$, $M$ and $Q$ so that $PM=MQ$. Prove that $AP+BQ>2CM$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/a8031007f5afc31a8b5cef98dd025474ac0351.png[/img] [b]7.5.[/b] Given $2n+1$ different objects. Prove that you can choose an odd number of objects from them in as many ways as an even number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].