A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. Each of the thirteen marked points is assigned a color, green or red. Prove that there are always three points of the same color, which are the vertices of an equilateral triangle.

Let $F$ be the collection of subsets of a set with $n$ elements such that no element of $F$ is a subset of another of its elements. Prove that $$|F|\le\binom n{\lfloor n/2\rfloor}.$$

Given a natural number $k> 1$. Prove that if through any edge of the graph $G$ passes less than $[e(k-1)! - 1]$ simple cycles, then the vertices of this graph can be colored with $k$ colors in the correct way.

Let's call $1 \times 2$ rectangle, which can be a rotated, a domino. Prove that there exists polygon, who can be covered by dominoes in exactly $2021$ different ways.

There are 101 coins in a circle, each weights 10g or 11g. Prove that there exists a coin such that the total weight of the $k{}$ coins to its left is equal to the total weight of the $k{}$ coins to its right where a) $k = 50$ and b) $k = 49$. [i]Alexandr Gribalko[/i]

For a finite non empty set of primes $P$, let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$. (i) Show that $|P|\le m(P)$, with equality if and only if $\min(P)>|P|$. (ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$. (The number $|P|$ is the size of set $P$) [i]Dan Schwarz, Romania[/i]

Peter wrote $100$ distinct integers on a board. Basil needs to fill the cells of a table $100\times{100}$ with integers so that the sum in each rectangle $1\times{3}$ (either vertical, or horizontal) is equal to one of the numbers written on the board. Find the greatest $n$ such that, regardless of numbers written by Peter, Basil can fill the table so that it would contain each of numbers $(1,2,...,n)$ at least once (and possibly some other integers).

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

In a $12\times 12$ square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones? [b]Note[/b]. Each diagonal is parallel to one of two main diagonals of the table and consists of $1,2\ldots,11$ or $12$ cells.

Barry has a standard die containing the numbers 1-6 on its faces. He rolls the die continuously, keeping track of the sum of the numbers he has rolled so far, starting from 0. Let $E_n$ be the expected number of time he needs to until his recorded sum is at least $n$. It turns out that there exist positive reals $a, b$ such that $$\lim_{n \rightarrow \infty} E_n - (an + b) = 0$$ Find $(a,b)$. [i]Proposed by Dilhan Salgado[/i]

Suppse that $X=\{1,2, \ldots, 2^{1996}-1\}$, prove that there exist a subset $A$ that satisfies these conditions: a) $1\in A$ and $2^{1996}-1\in A$; b) Every element of $A$ except $1$ is equal to the sum of two (possibly equal) elements from $A$; c) The maximum number of elements of $A$ is $2012$. [i]Romania[/i]

Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

Let $k$ be a positive integer. Prove that one can partition the set $\{ 0,1,2,3, \cdots ,2^{k+1}-1 \}$ into two disdinct subsets $\{ x_1,x_2, \cdots, x_{2k} \}$ and $\{ y_1, y_2, \cdots, y_{2k} \}$ such that $\sum_{i=1}^{2^k} x_i^m =\sum_{i=1}^{2^k} y_i^m$ for all $m \in \{ 1,2, \cdots, k \}$.

Prove that for all $n\in\mathbb{N}$ we have $\sum_{k=0}^n\dbinom {n}{k}^2=\dbinom {2n}{n}$.

Let an integer $ n > 3$. Denote the set $ T\equal{}\{1,2, \ldots,n\}.$ A subset S of T is called [i]wanting set[/i] if S has the property: There exists a positive integer $ c$ which is not greater than $ \frac {n}{2}$ such that $ |s_1 \minus{} s_2|\ne c$ for every pairs of arbitrary elements $ s_1,s_2\in S$. How many does a [i]wanting set[/i] have at most are there ?

Find the maximal count of shapes that can be placed over a chessboard with size $8\times8$ in such a way that no three shapes are not on two squares, lying next to each other by diagonal parallel $A1-H8$ ($A1$ is the lowest-bottom left corner of the chessboard, $H8$ is the highest-upper right corner of the chessboard). [i]V. Chukanov[/i]

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] What is $20 \div 2 - 0 \times 1 + 2 \times 5$? [b]p2.[/b] Today is Saturday, January $25$, $2020$. Exactly four hundred years from today, January $25$, $2420$, is again a Saturday. How many weekend days (Saturdays and Sundays) are in February, $2420$? (January has $31$ days and in year $2040$, February has $29$ days.) [b]p3.[/b] Given that there are four people sitting around a circular table, and two of them stand up, what is the probability that the two of them were originally sitting next to each other? [b]p4.[/b] What is the area of a triangle with side lengths $5$, $5$, and $6$? [b]p5.[/b] Six people go to OBA Noodles on Main Street. Each person has $1/2$ probability to order Duck Noodle Soup, $1/3$ probability to order OBA Ramen, and $1/6$ probability to order Kimchi Udon Soup. What is the probability that three people get Duck Noodle Soup, two people get OBA Ramen, and one person gets Kimchi Udon Soup? [b]p6.[/b] Among all positive integers $a$ and $b$ that satisfy $a^b = 64$, what is the minimum possible value of $a+b$? [b]p7.[/b] A positive integer $n$ is called trivial if its tens digit divides $n$. How many two-digit trivial numbers are there? [b]p8.[/b] Triangle $ABC$ has $AB = 5$, $BC = 13$, and $AC = 12$. Square $BCDE$ is constructed outside of the triangle. The perpendicular line from $A$ to side $DE$ cuts the square into two parts. What is the positive difference in their areas? [b]p9.[/b] In an increasing arithmetic sequence, the first, third, and ninth terms form an increasing geometric sequence (in that order). Given that the first term is $5$, find the sum of the first nine terms of the arithmetic sequence. [b]p10.[/b] Square $ABCD$ has side length $1$. Let points $C'$ and $D'$ be the reflections of points $C$ and $D$ over lines $AB$ and $BC$, respectively. Let P be the center of square $ABCD$. What is the area of the concave quadrilateral $PD'BC'$? [b]p11.[/b] How many four-digit palindromes are multiples of $7$? (A palindrome is a number which reads the same forwards and backwards.) [b]p12.[/b] Let $A$ and $B$ be positive integers such that the absolute value of the difference between the sum of the digits of $A$ and the sum of the digits of $(A + B)$ is $14$. What is the minimum possible value for $B$? [b]p13.[/b] Clark writes the following set of congruences: $x \equiv a$ (mod $6$), $x \equiv b$ (mod $10$), $x \equiv c$ (mod $15$), and he picks $a$, $b$, and $c$ to be three randomly chosen integers. What is the probability that a solution for $x$ exists? [b]p14.[/b] Vincent the bug is crawling on the real number line starting from $2020$. Each second, he may crawl from $x$ to $x - 1$, or teleport from $x$ to $\frac{x}{3}$ . What is the least number of seconds needed for Vincent to get to $0$? [b]p15.[/b] How many positive divisors of $2020$ do not also divide $1010$? [b]p16.[/b] A bishop is a piece in the game of chess that can move in any direction along a diagonal on which it stands. Two bishops attack each other if the two bishops lie on the same diagonal of a chessboard. Find the maximum number of bishops that can be placed on an $8\times 8$ chessboard such that no two bishops attack each other. [b]p17.[/b] Let $ABC$ be a right triangle with hypotenuse $20$ and perimeter $41$. What is the area of $ABC$? [b]p18.[/b] What is the remainder when $x^{19} + 2x^{18} + 3x^{17} +...+ 20$ is divided by $x^2 + 1$? [b]p19.[/b] Ben splits the integers from $1$ to $1000$ into $50$ groups of $20$ consecutive integers each, starting with $\{1, 2,...,20\}$. How many of these groups contain at least one perfect square? [b]p20.[/b] Trapezoid $ABCD$ with $AB$ parallel to $CD$ has $AB = 10$, $BC = 20$, $CD = 35$, and $AD = 15$. Let $AD$ and $BC$ intersect at $P$ and let $AC$ and $BD$ intersect at $Q$. Line $PQ$ intersects $AB$ at $R$. What is the length of $AR$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be any positive integer, and let $S(n)$ denote the number of permutations $\tau$ of $\{1,\dots,n\}$ such that $k^4+(\tau(k))^4$ is prime for all $k=1,\dots,n$. Show that $S(n)$ is always a square.

Let $S$ be the set of all n-tuples $(X_1,...,X_n)$ of subsets of the set $\{1,2,..,1000\}$, not necessarily different and not necessarily nonempty. For $a = (X_1,...,X_n)$ denote by $E(a)$ the number of elements of $X_1\cup ... \cup X_n$. Find an explicit formula for the sum $\sum_{a\in S} E(a)$

Consider $n\geq 3$ points in the plane, no three of which are collinear. For every convex polygon with vertices among the $n$ points, place $k\cdot 2^k$ coins in every one of its vertices, where $k$ is the number of points strictly in the interior of the polygon. Show that in total, no matter the configuration of the $n$ points, there are at most $n(n+1)\cdot 2^{n-3}$ placed coins. [i]Cristi Săvescu[/i]

Gary plays the following game with a fair $n$-sided die whose faces are labeled with the positive integers between $1$ and $n$, inclusive: if $n = 1$, he stops; otherwise he rolls the die, and starts over with a $k$-sided die, where $k$ is the number his $n$-sided die lands on. (In particular, if he gets $k = 1$, he will stop rolling the die.) If he starts out with a $6$-sided die, what is the expected number of rolls he makes?

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\tfrac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of his points against the other nine of the ten). What was the total number of players in the tournament?

Is it possible to fill an $n \times n$ table with the numbers $-1$, $0$ and $1$ so that all $2n$ sums in each column and each row are different? Solve the problem with a) $n = 5$; b) $n = 10$.