Given seven points on a plane, with no three points collinear. Prove that it is always possible to divide these points into the vertices of a triangle and a convex quadrilateral, with no shared parts between the two shapes.

Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened. Decide whether there exists a strategy for player $A$ to win in a finite number of moves.

Let us have $n$ ( $n>3$) balls with different rays. On each ball it is written an integer number. Determine the greatest natural number $d$ such that for any numbers written on the balls, we can always find at least 4 different ways to choose some balls with the sum of the numbers written on them divisible by $d$.

In every cell of a board $101 \times 101$ is written a positive integer. For any choice of $101$ cells from different rows and columns, their sum is divisible by $101$. Show that the number of ways to choose a cell from each row of the board, so that the total sum of the numbers in the chosen cells is divisible by $101$, is divisible by $101$.

$a)$ Prove that there exists $5$ nonnegative real numbers with sum equal to $1$, such that no matter how we arrange them on circle, two neighboring numbers exist with product not less than $\frac{1}{9}$ $a)$ Prove that for every $5$ nonnegative real numbers with sum equal to $1$, we can arrange them on circle, such that product of every two neighboring numbers is not greater than $\frac{1}{9}$

[b]p1.[/b] A triangle $ABC$ is drawn in the plane. A point $D$ is chosen inside the triangle. Show that the sum of distances $AD+BD+CD$ is less than the perimeter of the triangle. [b]p2.[/b] In a triangle $ABC$ the bisector of the angle $C$ intersects the side $AB$ at $M$, and the bisector of the angle $A$ intersects $CM$ at the point $T$. Suppose that the segments $CM$ and $AT$ divided the triangle $ABC$ into three isosceles triangles. Find the angles of the triangle $ABC$. [b]p3.[/b] You are given $100$ weights of masses $1, 2, 3,..., 99, 100$. Can one distribute them into $10$ piles having the following property: the heavier the pile, the fewer weights it contains? [b]p4.[/b] Each cell of a $10\times 10$ table contains a number. In each line the greatest number (or one of the largest, if more than one) is underscored, and in each column the smallest (or one of the smallest) is also underscored. It turned out that all of the underscored numbers are underscored exactly twice. Prove that all numbers stored in the table are equal to each other. [b]p5.[/b] Two stores have warehouses in which wheat is stored. There are $16$ more tons of wheat in the first warehouse than in the second. Every night exactly at midnight the owner of each store steals from his rival, taking a quarter of the wheat in his rival's warehouse and dragging it to his own. After $10$ days, the thieves are caught. Which warehouse has more wheat at this point and by how much? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Kevin has a square piece of paper with creases drawn to split the paper in half in both directions, and then each of the four small formed squares diagonal creases drawn, as shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/2/70d6c54e86856af3a977265a8054fd9b0444b0.png[/img] Find the sum of the corresponding numerical values of figures below that Kevin can create by folding the above piece of paper along the creases. (The figures are to scale.) Kevin cannot cut the paper or rip it in any way. [img]https://cdn.artofproblemsolving.com/attachments/a/c/e0e62a743c00d35b9e6e2f702106016b9e7872.png[/img]

$\definecolor{A}{RGB}{70,80,0}\color{A}\fbox{C4.}$ Show that for any positive integer $n \ge 3$ and some subset of $\lbrace{1, 2, . . . , n}\rbrace$ with size more than $\frac{n}2 + 1$, there exist three distinct elements $a, b, c$ in the subset such that $$\definecolor{A}{RGB}{255,70,255}\color{A} (ab)^2 + (bc)^2 + (ca)^2$$is a perfect square. [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1736[/color]

Two players $A$ and $B$ take turns playing the following game: There is a pile of $2003$ stones. In his first turn, $A$ selects a divisor of $2003$ and removes this number of stones from the pile. $B$ then chooses a divisor of the number of remaining stones, and removes that number of stones from the new pile, and so on. The player who has to remove the last stone loses. Show that one of the two players has a winning strategy and describe the strategy.

Let $n$ $\geq$ $2$ and $k$ $\geq$ $2$ be positive integers. A cat and a mouse are playing [i]Wim[/i], which is a stone removal game. The game starts with $n$ stones and they take turns removing stones, with the cat going first. On each turn they are allowed to remove $1$, $2$, $\dotsb$, or $k$ stones, and the player who cannot remove any stones on their turn loses. \\\\ A raccoon finds Wim very boring and creates [i]Wim 2[/i], which is Wim but with the following additional rule: [i]You cannot remove the same number of stones that your opponent removed on the previous turn[/i]. \\\\Find all values of $k$ such that for every $n$, the cat has a winning strategy in Wim if and only if it has a winning strategy in Wim 2.

In a plane a finite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of different colors. What is the smallest number of circles that requires four colors?

Karlsson-on-the-Roof has $1000$ jars of jam. The jars are not necessarily identical; each contains no more than $\dfrac{1}{100}$-th of the total amount of the jam. Every morning, Karlsson chooses any $100$ jars and eats the same amount of the jam from each of them. Prove that Karlsson can eat all the jam. [i](8 points)[/i]

There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define: $(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$; $(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities; $(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities. Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$ Proposed by David-Andrei Anghel, Romania.

There are $2022$ equally spaced points on a circular track $\gamma$ of circumference $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle. Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labellings of the $2022$ points. [i]Kevin Cong[/i]

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

A cross with length $p$ (or [i]p-cross[/i] for short) will be called the figure formed by a unit square and 4 rectangles $p-1$ x $1$ on its sides. What’s the least amount of colors one has to use to color the cells of an infinite table, so that each [i]p-cross[/i] on it covers cells, no two of which are in the same color?

Two permutations of $1,\ldots, n$ are written on the board: $a_1,\ldots,a_n$ $b_1,\ldots,b_n$ A move consists of one of the following two operations: 1) Change the first row to $b_{a_1},\ldots,b_{a_n}$ 2) Change the second row to $a_{b_1},\ldots,a_{b_n}$ The starting position is: $2134\ldots n$ $234\ldots n1$ Is it possible by finitely many moves to get: $2314\ldots n$ $234 \ldots n1$? [i]D. Zmiaikou[/i]

A sequence $a_1,\ldots,a_n$ of positive integers is given. For each $l$ from $1$ to $n-1$ the array $(gcd(a_1,a_{1+l}),\ldots,gcd(a_n,a_{n+l}))$ is considered, where indices are taken modulo $n$. It turned out that all this arrays consist of the same $n$ pairwise distinct numbers and differ only,possibly, by their order. Can $n$ be a) $21$ b) $2021$

$N$ teams participated in a national basketball championship in which every two teams played exactly one game. Of the $N$ teams, $251$ are from California. It turned out that a Californian team Alcatraz is the unique Californian champion (Alcatraz has won more games against Californian teams than any other team from California). However, Alcatraz ended up being the unique loser of the tournament because it lost more games than any other team in the nation! What is the smallest possible value for $N$?

On the square, $1,995$ soldiers lined up in a column, and some of them stood correctly, and some turned backwards. Sergeant Smith remembers only the command "as". With this command, each soldier who sees an even number of faces facing him turns $180^o$, while the rest remain stationary. All movements on command are performed simultaneously. Prove that the sergeant can orient all the soldiers in one direction.

In a circle, $15$ equally spaced points are drawn and arbitrary triangles are formed connecting $3$ of these points. How many non-congruent triangles can be drawn?

Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles. [i]UK - Sahl Khan[/i]

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \(0\le n\le 52\), and cuts the deck into a pile with the top \(n\) cards and a pile with the remaining \(52-n\). She then gives both piles to the volunteer, who riffles them together and hands the deck back to her face down. (Thus, in the resulting deck, the cards that were in the deck of size \(n\) appear in order, as do the cards that were in the deck of size \(52-n\).) Alice then flips the cards over one-by-one from the top. Before flipping over each card, she may choose to guess the color of the card she is about to flip over. She stops if she guesses incorrectly. What is the maximum number of correct guesses she can guarantee? [i]Proposed by Espen Slettnes[/i]