Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

An [i]augmentation[/i] on a graph $G$ is defined as doing the following: - Take some set $D$ of vertices in $G$, and duplicate each vertex $v_i \in D$ to create a new vertex $v_i'$. - If there's an edge between a pair of vertices $v_i, v_j \in D$, create an edge between vertices $v_i'$ and $v_j'$. If there's an edge between a pair of vertices $v_i \in D$, $v_j \notin D$, you can choose to create an edge between $v_i'$ and $v_j$ but do not have to. A graph is called [i]reachable[/i] from $G$ if it can be created through some sequence of augmentations on $G$. Some graph $H$ has $n$ vertices and satisfies that both $H$ and the complement of $H$ are reachable from a complete graph of $2021$ vertices. If the maximum and minimum values of $n$ are $M$ and $m$, find $M+m$. [i]Proposed by Oliver Hayman[/i]

A $ 9 \times 9$ square is divided into unit squares. Is it possible to fill each unit square with a number $ 1, 2,..., 9$ in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?

[b]p1.[/b] Suppose that $a \oplus b = ab - a - b$. Find the value of $$((1 \oplus 2) \oplus (3 \oplus 4)) \oplus 5.$$ [b]p2.[/b] Neethin scores a $59$ on his number theory test. He proceeds to score a $17$, $23$, and $34$ on the next three tests. What score must he achieve on his next test to earn an overall average of $60$ across all five tests? [b]p3.[/b] Consider a triangle with side lengths $28$ and $39$. Find the number of possible integer lengths of the third side. [b]p4.[/b] Nithin is thinking of a number. He says that it is an odd two digit number where both of its digits are prime, and that the number is divisible by the sum of its digits. What is the sum of all possible numbers Nithin might be thinking of? [b]p5.[/b] Dora sees a fire burning on the dance floor. She calls her friends to warn them to stay away. During the first pminute Dora calls Poonam and Serena. During the second minute, Poonam and Serena call two more friends each, and so does Dora. This process continues, with each person calling two new friends every minute. How many total people would know of the fire after $6$ minutes? [b]p6.[/b] Charlotte writes all the positive integers $n$ that leave a remainder of $2$ when $2018$ is divided by $n$. What is the sum of the numbers that she writes? [b]p7.[/b] Consider the following grid. Stefan the bug starts from the origin, and can move either to the right, diagonally in the positive direction, or upwards. In how many ways can he reach $(5, 5)$? [img]https://cdn.artofproblemsolving.com/attachments/9/9/b9fdfdf604762ec529a1b90d663e289b36b3f2.png[/img] [b]p8.[/b] Let $a, b, c$ be positive numbers where $a^2 + b^2 + c^2 = 63$ and $2a + 3b + 6c = 21\sqrt7$. Find $\left( \frac{a}{c}\right)^{\frac{a}{b}} $. [b]p9.[/b] What is the sum of the distinct prime factors of $12^5 + 12^4 + 1$? [b]p10.[/b] Allen starts writing all permutations of the numbers $1$, $2$, $3$, $4$, $5$, $6$ $7$, $8$, $9$, $10$ on a blackboard. At one point he writes the permutation $9$, $4$, $3$, $1$, $2$, $5$, $6$, $7$, $8$, $10$. David points at the permutation and observes that for any two consecutive integers $i$ and $i+1$, all integers that appear in between these two integers in the permutation are all less than $i$. For example, $4$ and $5$ have only the numbers $3$, $1$, $2$ in between them. How many of the $10!$ permutations on the board satisfy this property that David observes? [b]p11.[/b] (Estimation) How many positive integers less than $2018$ can be expressed as the sum of $3$ square numbers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Assume integer $m \geq 2.$ There are $3m$ people in a meeting, any two of them either shake hands with each other once or not.We call the meeting "$n$-interesting", only if there exists $n(n\leq 3m-1)$ people of them, the time everyone of whom shakes hands with other $3m-1$ people is exactly $1,2,\cdots,n,$ respectively. If in any "$n$-interesting" meeting, there exists $3$ people of them who shake hands with each other, find the minimum value of $n.$

Benji has a $2\times 2$ grid, which he proceeds to place chips on. One by one, he places a chip on one of the unit squares of the grid at random. However, if at any point there is more than one chip on the same square, Benji moves two chips on that square to the two adjacent squares, which he calls a chip-fire. He keeps adding chips until there is an infinite loop of chip-fires. What is the expected number of chips that will be added to the board?

i) Show there exist (not necessarily distinct) non-negative real numbers $a_1,a_2,\ldots,a_{10}$; $b_1,b_2,\ldots,b_{10}$, with $a_k+b_k \leq 4$ for all $1\leq k \leq 10$, such that $\max\{|a_i-a_j|, |b_i-b_j|\} \geq \dfrac{4}{3} > 1$ for all $1\leq i < j \leq 10$. ii) Prove for any (not necessarily distinct) non-negative real numbers $a_1,a_2,\ldots,a_{11}$; $b_1,b_2,\ldots,b_{11}$, with $a_k+b_k \leq 4$ for all $1\leq k \leq 11$, there exist $1\leq i < j \leq 11$ such that $\max\{|a_i-a_j|, |b_i-b_j|\} \leq 1$. ([i]Dan Schwarz[/i])

Ten evenly spaced vertical lines in the plane are labeled $\ell_1,\ell_2, \ldots,\ell_{10}$ from left to right. A set $\{a,b,c,d\}$ of four distinct integers $a,b,c,d \in \{1,2,\ldots,10\}$ is [i]squarish[/i] if some square has one vertex on each of the lines $\ell_a,\ell_b,\ell_c,$ and $\ell_d.$ Find the number of squarish sets.

Ten monkeys have 60 bananas. Each monkey has at least one banana and any two monkeys have different amounts of bananas. Prove that any six monkeys can distribute their bananas between others such that all 4 remaining monkeys have the same amount of bananas. [i]A. Voidelevich[/i]

The plane is painted in red or blue color. Prove that you have a rectangle with the corners of the same color.

Two students $A$ and $B$ play a game on a $20 \text{ x } 20$ chessboard. It is known that two squares are said to be [i]adjacent[/i] if the two squares have a common side. At the beginning, there is a chess piece in a certain square of the chessboard. Given that $A$ will be the first one to move the chess piece, $A$ and $B$ will alternately move this chess piece to an adjacent square. Also, the common side of any pair of adjacent squares can only be passed once. If the opponent cannot move anymore, then he will be declared the winner (to clarify since the wording wasn’t that good, you lose if you can’t move). Who among $A$ and $B$ has a winning strategy? Justify your claim.

[b]p1.[/b] Compute the value of $N$, where $$N = 818^3 - 6 \cdot 818^2 \cdot 209 + 12 \cdot 818 \cdot 209^2 - 8 \cdot 209^3$$ [b]p2.[/b] Suppose $x \le 2019$ is a positive integer that is divisible by $2$ and $5$, but not $3$. If $7$ is one of the digits in $x$, how many possible values of $x$ are there? [b]p3.[/b] Find all non-negative integer solutions $(a,b)$ to the equation $$b^2 + b + 1 = a^2.$$ [b]p4.[/b] Compute the remainder when $\sum^{2019}_{n=1} n^4$ is divided by $53$. [b]p5.[/b] Let $ABC$ be an equilateral triangle and $CDEF$ a square such that $E$ lies on segment $AB$ and $F$ on segment $BC$. If the perimeter of the square is equal to $4$, what is the area of triangle $ABC$? [img]https://cdn.artofproblemsolving.com/attachments/1/6/52d9ef7032c2fadd4f97d7c0ea051b3766b584.png[/img] [b]p6.[/b] $$S = \frac{4}{1\times 2\times 3}+\frac{5}{2\times 3\times 4} +\frac{6}{3\times 4\times 5}+ ... +\frac{101}{98\times 99\times 100}$$ Let $T = \frac54 - S$. If $T = \frac{m}{n}$ , where $m$ and $n$ are relatively prime integers, find the value of $m + n$. [b]p7.[/b] Find the sum of $$\sum^{2019}_{i=0}\frac{2^i}{2^i + 2^{2019-i}}$$ [b]p8.[/b] Let $A$ and $B$ be two points in the Cartesian plane such that $A$ lies on the line $y = 12$, and $B$ lies on the line $y = 3$. Let $C_1$, $C_2$ be two distinct circles that intersect both $A$ and $B$ and are tangent to the $x$-axis at $P$ and $Q$, respectively. If $PQ = 420$, determine the length of $AB$. [b]p9.[/b] Zion has an average $2$ out of $3$ hit rate for $2$-pointers and $1$ out of $3$ hit rate for $3$-pointers. In a recent basketball match, Zion scored $18$ points without missing a shot, and all the points came from $2$ or $3$-pointers. What is the probability that all his shots were $3$-pointers? [b]p10.[/b] Let $S = \{1,2, 3,..., 2019\}$. Find the number of non-constant functions $f : S \to S$ such that $$f(k) = f(f(k + 1)) \le f(k + 1) \,\,\,\, for \,\,\,\, all \,\,\,\, 1 \le k \le 2018.$$ Express your answer in the form ${m \choose n}$, where $m$ and $n$ are integers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Maria has a $6 \times 5$ board with some shaded squares, as in the figure. She writes, in some order, the digits $1, 2, 3, 4$ and $5$ in the first row and then completes the board as follows: look at the number written in the shaded box and write the number that occupies the position indicated by the box shaded as the last number in the next row, and repeat the other numbers in the first four squares, following the same order as in the previous row. For example, if you wrote $2, 3, 4, 1, 5$ in the first row, then since $4$ is in the shaded box, the number that occupies the fourth place $(1)$ is written in the last box of the second row and completes it with the remaining numbers in the order in which. They were. She remains: $2, 3, 4, 5, 1$. Then, to complete the third row, as in the shaded box is $3$, the number located in the third place $(4)$ writes it in the last box and gets $2, 3, 5, 1, 4$. Following in the same way, he gets the board of the figure. Show a way to locate the numbers in the first row to get the numbers $2, 4, 5, 1, 3$ in the last row.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Tsrutsuna starts in the bottom left cell of a 7 × 7 square table, while Tsuna is in the upper right cell. The center cell of the table contains cheese. Tsrutsuna wants to reach Tsuna and bring a piece of cheese with him. From a cell Tsrutsuna can only move to the right or the top neighboring cell. Determine the number of different paths Tsrutsuna can take from the lower left cell to the upper right cell, such that he passes through the center cell. [i]Proposed by Giorgi Arabidze, Georgia[/i]

Consider a cross like in the figure but with size $2021$. Every square have a $+1$. Every minute we select a sub-cross of size $3$ and multiply their squares by $-1$. It´s posible achieve that all the squares of the cross with size $2021$ have a $-1$?

Let $n$ be a positive integer. We want to make up a collection of cards with the following properties: 1. each card has a number of the form $m!$ written on it, where $m$ is a positive integer. 2. for any positive integer $ t \le n!$, we can select some card(s) from this collection such that the sum of the number(s) on the selected card(s) is $t$. Determine the smallest possible number of cards needed in this collection.

All points in the plane that have both integer coordinates are painted, using the colors red, green, and yellow. If the points are painted so that there is at least one point of each color. Prove that there are always three points $X$, $Y$ and $Z$ of different colors, such that $\angle XYZ = 45^{\circ} $

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

Given is a convex octagon $A_1A_2 \ldots A_8$. Given a triangulation $T$, one can take two triangles $\triangle A_iA_jA_k$ and $\triangle A_iA_kA_l$ and replace them with $\triangle A_iA_jA_l$ and $\triangle A_jA_lA_k$. Find the minimal number of operations $k$ we have to do so that for any pair of triangulations $T_1, T_2$, we can reach $T_2$ from $T_1$ using at most $k$ operations.

An antelope is a chess piece which moves similarly to the knight: two cells $(x_1,y_1)$ and $(x_2,y_2)$ are joined by an antelope move if and only if \[ \{|x_1-x_2|,|y_1-y_2|\}=\{3,4\}.\] The numbers from $1$ to $10^{12}$ are placed in the cells of a $10^6\times 10^6$ grid. Let $D$ be the set of all absolute differences of the form $|a-b|$, where $a$ and $b$ are joined by an antelope move in the arrangement. How many arrangements are there such that $D$ contains exactly four elements? Proposed by [i]Nikolai Beluhov[/i], Bulgaria

In a circle there are $2019$ plates, on each lies one cake. Petya and Vasya are playing a game. In one move, Petya points at a cake and calls number from $1$ to $16$, and Vasya moves the specified cake to the specified number of check clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants at least some $k$ pastries to accumulate on one of the plates and Vasya wants to stop him. What is the largest $k$ Petya can succeed?

Let $P_1,P_2,\ldots ,P_n$, be infinite arithmetic progressions of positive integers, of differences $d_1,d_2,\ldots ,d_n$, respectively. Prove that if every positive integer appears in at least one of the $n$ progressions then one of the differences $d_i$ divides the least common multiple of the remaining $n-1$ differences. Note: $P_i=\left \{ a_i,a_i+d_i,a_i+2d_i,a_i+3d_i,a_i+4d_i,\cdots \right \}$ with $ a_i$ and $d_i$ positive integers.

[b]p33.[/b] Lines $\ell_1$ and $\ell_2$ have slopes $m_1$ and $m_2$ such that $0 < m_2 < m_1$. $\ell'_1$ and $\ell'_2$ are the reflections of $\ell_1$ and $\ell_2$ about the line $\ell_3$ defined by $y = x$. Let $A = \ell_1 \cap \ell_2 = (5, 4)$, $B = \ell_1 \cap \ell_3$, $C = \ell'_1 \cap \ell'_2$ and $D = \ell_2 \cap \ell_3$. If $\frac{4-5m_1}{-5-4m_1} = m_2$ and $\frac{(1+m^2_1)(1+m^2_2)}{(1-m_1)^2(1-m_2)^2} = 41$, compute the area of quadrilateral $ABCD$. [b]p34.[/b] Suppose $S(m, n) = \sum^m_{i=1}(-1)^ii^n$. Compute the remainder when $S(2020, 4)$ is divided by $S(1010, 2)$. [b]p35.[/b] Let $N$ be the number of ways to place the numbers $1, 2, ..., 12$ on a circle such that every pair of adjacent numbers has greatest common divisor $1$. What is $N/144$? (Arrangements that can be rotated to yield each other are the same). [b]p36.[/b] Compute the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{{2n \choose 2}} =\frac{1}{{2 \choose 2}} - \frac{1}{{4 \choose 2}} +\frac{1}{{6 \choose 2}} -\frac{1}{{8 \choose 2}} -\frac{1}{{10 \choose 2}}+\frac{1}{{12 \choose 2}} +...$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A coin is tossed 9 times. Hence $2^{9}$ different outcomes are possible. In how many cases 2 consecutive heads does not appear? [list=1] [*] 34 [*] 55 [*] 89 [*] None of these [/list]