A triangle is given on a sphere of radius $1$, the sides of which are arcs of three different circles of radius $1$ centered in the center of a sphere having less than $\pi$ in length and an area equal to a quarter of the area of the sphere. Prove that four copies of such a triangle can cover the entire sphere. A. Zaslavsky

Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.

For a set $S$ of nonnegative integers, let $r_S(n)$ denote the number of ordered pairs $(s_1, s_2)$ such that $s_1 \in S$, $s_2 \in S$, $s_1 \neq s_2$, and $s_1 + s_2 = n$. Is it possible to partition the nonnegative integers into two sets $A$ and $B$ in such a way that $r_A(n) = r_B(n)$ for all $n$?

$S,T$ are two trees without vertices of degree 2. To each edge is associated a positive number which is called length of this edge. Distance between two arbitrary vertices $v,w$ in this graph is defined by sum of length of all edges in the path between $v$ and $w$. Let $f$ be a bijective function from leaves of $S$ to leaves of $T$, such that for each two leaves $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $f(u), f(v)$ in $T$. Prove that there is a bijective function $g$ from vertices of $S$ to vertices of $T$ such that for each two vertices $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $g(u)$ and $g(v)$ in $T$.

Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points. [b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$. [b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.

The numbers $1, 2, . . . , 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $k < 10$, there exists an integer $k' > k$ such that there is at most one number between $k$ and $k'$ in the circle. If $p$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.

Several pupils with different heights are standing in a row. If they were arranged according to their heights, such that the highest would stand on the right, then each pupil would move for at most 8 positions. Prove that every pupil has no more than 8 pupils lower then him on his right.

Each edge and each diagonal of the convex $ n$-gon $ (n\geq 3)$ is colored in red or blue. Prove that the vertices of the $ n$-gon can be labeled as $ A_1,A_2,...,A_n$ in such a way that one of the following two conditions is satisfied: (a) all segments $ A_1A_2,A_2A_3,...,A_{n\minus{}1}A_n,A_nA_1$ are of the same colour. (b) there exists a number $ k, 1<k<n$ such that the segments $ A_1A_2,A_2A_3,...,A_{k\minus{}1}A_k$ are blue, and the segments $ A_kA_{k\plus{}1},...,A_{n\minus{}1}A_n,A_nA_1$ are red.

Prove that however we choose the majority of numbers among an even number of the first consecutive natural numbers, there will be two numbers among this choosing whose sum is a prime.

For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$. Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.

Let $N$ be the number of tuples $(a_1, a_2,..., a_{150})$ satisfying: $\bullet$ $a_i \in \{2, 3, 5, 7, 11\}$ for all $1 \le i \le 99$. $\bullet$ $a_i \in \{2, 4, 6, 8\}$ for all $100 \le i \le 150$. $\bullet$ $\sum^{150}_{i=1}a_i$ is divisible by $8$. Compute the last three digits of $N$.

nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square. [i]Proposed by Mihir Singhal and Michael Kural[/i]

For covering the floor of a rectangular room rectangular tiles of sizes $2 \times 2$ and $4 \times 1$ were used. Show that it's not possible to cover the floor if there is one plate less of one type and one more of the other type.

A sequence $P_1, \dots, P_n$ of points in the plane (not necessarily diferent) is [i]carioca[/i] if there exists a permutation $a_1, \dots, a_n$ of the numbers $1, \dots, n$ for which the segments $$P_{a_1}P_{a_2}, P_{a_2}P_{a_3}, \dots, P_{a_n}P_{a_1}$$ are all of the same length. Determine the greatest number $k$ such that for any sequence of $k$ points in the plane, $2023-k$ points can be added so that the sequence of $2023$ points is [i]carioca[/i].

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

A permutation of $\{1, 2 \cdots, n \}$ is [i]magic[/i] if each element $k$ of it has at least $\left\lfloor \frac{k}{2} \right\rfloor$ numbers less to it at the left. For each $n$ find the number of [i]magical[/i] permutations.

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

A $4\times4$ grid has its 16 cells colored arbitrarily in three colors. A [i]swap[/i] is an exchange between the colors of two cells. Prove or disprove that it always takes at most three swaps to produce a line of symmetry, regardless of the grid's initial coloring. [i]Proposed by Matthew Babbitt[/i]

A group of $ 100$ students numbered $ 1$ through $ 100$ are playing the following game. The judge writes the numbers $ 1$, $ 2$, $ \ldots$, $ 100$ on $ 100$ cards, places them on the table in an arbitrary order and turns them over. The students $ 1$ to $ 100$ enter the room one by one, and each of them flips $ 50$ of the cards. If among the cards flipped by student $ j$ there is card $ j$, he gains one point. The flipped cards are then turned over again. The students cannot communicate during the game nor can they see the cards flipped by other students. The group wins the game if each student gains a point. Is there a strategy giving the group more than $ 1$ percent of chance to win?

$2000$ people are standing on a line. Each one of them is either a [i]liar[/i], who will always lie, or a [i]truth-teller[/i], who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line.

[u]Round 1[/u] [b]p1.[/b] Ash is running around town catching Pokémon. Each day, he may add $3, 4$, or $5$ Pokémon to his collection, but he can never add the same number of Pokémon on two consecutive days. What is the smallest number of days it could take for him to collect exactly $100$ Pokémon? [b]p2.[/b] Jack and Jill have ten buckets. One bucket can hold up to $1$ gallon of water, another can hold up to $2$ gallons, and so on, with the largest able to hold up to $10$ gallons. The ten buckets are arranged in a line as shown below. Jack and Jill can pour some amount of water into each bucket, but no bucket can have less water than the one to its left. Is it possible that together, the ten buckets can hold 36 gallons of water? [img]https://cdn.artofproblemsolving.com/attachments/f/8/0b6524bebe8fe859fe7b1bc887ac786106fc17.png[/img] [b]p3.[/b] There are $2023$ knights and liars standing in a row. Knights always tell the truth and liars always lie. Each of them says, “the number of liars to the left of me is greater than the number of knights to the right.” How many liars are there? [b]p4.[/b] Camila has a deck of $101$ cards numbered $1, 2, ..., 101$. She starts with $50$ random cards in her hand and the rest on a table with the numbers visible. In an exchange, she replaces all $50$ cards in her hand with her choice of $50$ of the $51$ cards from the table. Show that Camila can make at most 50 exchanges and end up with cards $1, 2, ..., 50$. [img]https://cdn.artofproblemsolving.com/attachments/0/6/c89e65118764f3b593da45264bfd0d89e95067.png[/img] [b]p5.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [u]Round 2[/u] [b]p6.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company will lay lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses. The Edison lighting company will hang strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [b]p7.[/b] You are given a sequence of $16$ digits. Is it always possible to select one or more digits in a row, so that multiplying them results in a square number? [img]https://cdn.artofproblemsolving.com/attachments/d/1/f4fcda2e1e6d4a1f3a56cd1a04029dffcd3529.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $n$ teams in a football league. During a championship, every two teams play exactly one match, but no team can play more than one match in a week. At least, how many weeks are necessary for the championship to be held? Give an schedule for such a championship.

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]

$N \ge 3$ different points are marked on the plane. It is known that among pairwise distances between marked points there are not more than $n$ different distances. Prove that $N \le (n + 1)^2$.

We will say that a subset $A$ of the set of non-negative integers is $cool$, if there exist an integer $k$, such that for every integer $n\geq k$ there exists exactly one pair of integers $a>b$ from $A$ such that $n=a+b$. Decide, if there exists a $cool$ set.