In a class with $n$ students in the span of $k$ days, each day are chosen three to be tested. Each two students can be taken in such triple only once. Prove that for the greatest $k$ satisfying these conditions, the following inequalities are true: $\frac{n(n-3)}{6}\leq k\leq \frac{n(n-1)}{6}$.

Each cell of the board with $2021$ rows and $2022$ columns contains exactly one of the three letters $T$, $M$, and $O$ in a way that satisfies each of the following conditions: [list] [*] In total, each letter appears exactly $2021\times 674$ of times on the board. [*] There are no two squares that share a common side and contain the same letter. [*] Any $2\times 2$ square contains all three letters $T$, $M$, and $O$. [/list] Prove that each letter $T$, $M$, and $O$ appears exactly $674$ times on every row.

Let $k$ and $s$ be positive integers such that $s<(2k + 1)^2$. Initially, one cell out of an $n \times n$ grid is coloured green. On each turn, we pick some green cell $c$ and colour green some $s$ out of the $(2k + 1)^2$ cells in the $(2k + 1) \times (2k + 1)$ square centred at $c$. No cell may be coloured green twice. We say that $s$ is $k-sparse$ if there exists some positive number $C$ such that, for every positive integer $n$, the total number of green cells after any number of turns is always going to be at most $Cn$. Find, in terms of $k$, the least $k$-sparse integer $s$. [I]Proposed by Nikolai Beluhov.[/i]

In a triangle $T$, all the angles are less than $90^o$, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the distance from $p$ to $h$ is less than or equal to $s/\sqrt3$.

Let $n$ be a positive integer. Chef Kao has $n$ different flavors of ice cream. He wants to serve one small cup and one large cup for each flavor. He arranges the $2n$ ice cream cups into two rows of $n$ cups on a tray. He wants the tray to be colorful, so he arranges the ice cream cups with the following conditions: [list] [*]each row contains all ice cream flavors, and [*]each column has different sizes of ice cream cup. [/list]Determine the number of ways that Chef Kao can arrange cups of ice cream with the above conditions.

Let $V$ be the set of vertices of a regular $21$-gon. Given a non-empty subset $U$ of $V$ , let $m(U)$ be the number of distinct lengths that occur between two distinct vertices in $U$. What is the maximum value of $\frac{m(U)}{|U|}$ as $U$ varies over all non-empty subsets of $V$ ?

There are eight different symbols designed on $n\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\leq k\leq 7)$ the number of shirts containing at least one of the $k$ symbols is even. Determine the value of $n$.

The diagram below shows four congruent squares and some of their diagonals. Let $T$ be the number of triangles and $R$ be the number of rectangles that appear in the diagram. Find $T + R$. [img]https://cdn.artofproblemsolving.com/attachments/1/5/f756bbe67c09c19e811011cb6b18d0ff44be8b.png[/img]

In a multiple choice test there were 4 questions and 3 possible answers for each question. A group of students was tested and it turned out that for any three of them there was a question which the three students answered differently. What is the maximum number of students tested?

Let $D$ be the set of all pairs $(i,j)$, $1\le i,j\le n$. Prove there exists a subset $S \subset D$, with $|S|\ge\left \lfloor\frac{3n(n+1)}{5}\right \rfloor$, such that for any $(x_1,y_1), (x_2,y_2) \in S$ we have $(x_1+x_2,y_1+y_2) \not \in S$. (Peter Cameron)

The professor tells Peter the product of two positive integers and Sam their sum. At first, nobody of them knows the number of the other. One of them says: [i]You can't possibly guess my number[/i]. Then the other says: [i]You are wrong, the number is 136[/i]. Which number did the professor tell them respectively? Give reasons for your claim.

In Hidro planet were living $2008^2$ hydras some time ago. One of them had 1 tentacle, other 2 and so on to the last with $2008^2$ tentacles. Let $t(H)$ be the number of tentacles of hydra $H$. The pairing of $H_1$ and $H_2$ (where $t(H_1) < t(H_2)$) is a new hydra with $t(H_2)-t(H_1)+8$ tentacles, in case of $t(H_1)=t(H_2)$ they die. An expedition found in Hidro the last hydra with 23 tentacles. That could be true ?

A teacher gives a multiple choice test to $15$ students and that each student answered each question. Each question had $5$ choices, but remarkably, no pair of students had more than $2$ answers in common. What is the maximum number of questions that could have been on the quiz?

There is a game. The magician let the participant think up a positive integer (at least two digits). For example, an integer $ \displaystyle\overline{a_1a_2 \cdots a_n}$ is rearranged as $ \overline{a_{i_1}a_{i_2} \cdots a_{i_n}}$, that is, $ i_1, i_2, \cdots, i_n$ is a permutation of $ 1,2, \cdots, n$. Then we get $ n!\minus{}1$ integers. The participant is asked to calculate the sum of the $ n!\minus{}1$ numbers, then tell the magician the sum $ S$. The magician claims to be able to know the original number when he is told the sum $ S$. Try to decide whether the magician can be successful or not.

Régis, Ed and Rafael are at the IMO. They are going to play a game in Bath, and there are $2^n$ houses in the city. Régis and Ed will team up against Rafael. The game operates as follows: First, Régis and Ed think on a strategy and then let Rafael know it. After this, Régis and Ed no longer communicate, and the game begins. Rafael decides on an order to visit the houses and then starts taking Régis to them in that order. At each house, except for the last one, Régis choose a number between $1$ and $n$ and places it in the house. In the last house, Rafael chooses a number from $1$ to $n$ and places it there. Afterwards, Ed sees all the houses and the numbers in them, and he must guess in which house Rafael placed the number. Ed is allowed $k$ guesses. What is the smallest $k$ for which there exists a strategy for Ed and Régis to ensure that Ed correctly guess the house where Rafael placed the number?

Let $s,t$ be given nonzero integers, $(x,y)$ be any (ordered) pair of integers. A sequence of moves is performed as follows: per move $(x,y)$ changes to $(x+t, y-s)$. The pair (x,y) is said to be [i]good [/i] if after some (may be, zero) number of moves described a pair of integers arises that are not relatively prime. a) Determine whether $(s,t)$ is itself a good pair; bj Prove that for any nonzero $s$ and $t$ there is a pair $(x,y)$ which is not good.

[b]7.1[/b] Given a convex $n$-gon all of whose angles are obtuse. Prove that the sum of the lengths of the diagonals in it is greater than the sum of the lengths of the sides. [b]7.2[/b] Find all integer values for $x$ and $y$ such that $x^4 + 4y^4$ is a prime number[b]. (typo corrected)[/b] [b]7.3.[/b] Given a triangle $ABC$. Parallelograms $ABKL$, $BCMN$ and $ACFG$ are constructed on the sides, Prove that the segments $KN$, $MF$ and $GL$ can form a triangle. [img]https://cdn.artofproblemsolving.com/attachments/a/f/7a0264b62754fafe4d559dea85c67c842011fc.png[/img] [b]7.4 / 6.2[/b] Prove that a $10 \times 10$ chessboard cannot be covered with $ 25$ figures like [img]https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png[/img]. [b]7.5[/b] Find the greatest number of different natural numbers, each of which is less than $50$, and every two of which are coprime. [b]7.6.[/b] Given a triangle $ABC$.$ D$ and $E$ are the midpoints of the sides $AB$ and $BC$. Point$ M$ lies on $AC$ , $ME > EC$. Prove that $MD < AD$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/1dd901e0121e5c75a4039d21b954beb43dc547.png[/img] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

Prove that the number $ (\underbrace{9999 \dots 99}_{2005}) ^{2009}$ can be obtained by erasing some digits of $ (\underbrace{9999 \dots 99}_{2008}) ^{2009}$ (both in decimal representation). [i]Yudi Satria, Jakarta[/i]

A finite set $M$ of real numbers has the following properties: $M$ has at least $4$ elements, and there exists a bijective function $f:M\to M$, different from the identity, such that $ab\leq f(a)f(b)$ for all $a\neq b\in M.$ Prove that the sum of the elements of $M$ is $0.$

[u]Round 1[/u] [b]p1.[/b] At a fortune-telling exam, $13$ witches are sitting in a circle. To pass the exam, a witch must correctly predict, for everybody except herself and her two neighbors, whether they will pass or fail. Each witch predicts that each of the $10$ witches she is asked about will fail. How many witches could pass? [b]p2.[/b] Out of $152$ coins, $7$ are counterfeit. All counterfeit coins have the same weight, and all real coins have the same weight, but counterfeit coins are lighter than real coins. How can you find $19$ real coins if you are allowed to use a balance scale three times? [b]p3.[/b] The digits of a number $N$ increase from left to right. What could the sum of the digits of $9 \times N$ be? [b]p4.[/b] The sides and diagonals of a pentagon are colored either blue or red. You can choose three vertices and flip the colors of all three lines that join them. Can every possible coloring be turned all blue by a sequence of such moves? [img]https://cdn.artofproblemsolving.com/attachments/5/a/644aa7dd995681fc1c813b41269f904283997b.png[/img] [b]p5.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake and call that number $N$. Pick up the stack of the top $N$ pancakes and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [u]Round 2[/u] [b]p6.[/b] A circus owner will arrange $100$ fleas on a long string of beads, each flea on her own bead. Once arranged, the fleas start jumping using the following rules. Every second, each flea chooses the closest bead occupied by one or more of the other fleas, and then all fleas jump simultaneously to their chosen beads. If there are two places where a flea could jump, she jumps to the right. At the start, the circus owner arranged the fleas so that, after some time, they all gather on just two beads. What is the shortest amount of time it could take for this to happen? [b]p7.[/b] The faraway land of Noetheria has $2016$ cities. There is a nonstop flight between every pair of cities. The price of a nonstop ticket is the same in both directions, but flights between different pairs of cities have different prices. Prove that you can plan a route of $2015$ consecutive flights so that each flight is cheaper than the previous one. It is permissible to visit the same city several times along the way. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.

Shiori places seven books, numbered from $1$ to $7$, on a bookshelf in some order. Later, she discovers that she can place two dividers between the books, separating the books into left, middle, and right sections, such that: $\bullet$ The left section is numbered in increasing order from left to right, or has at most one book. $\bullet$ The middle section is numbered in decreasing order from left to right, or has at most one book. $\bullet$ The right section is numbered in increasing order from left to right, or has at most one book. In how many possible orderings could Shiori have placed the books? For example, $(2, 3, 5, 4, 1, 6, 7)$ and $(2, 3, 4, 1, 5, 6, 7)$ are possible orderings with the partitions $2, 3, 5|4, 1|6, 7$ and $2, 3, 4|1|5, 6, 7$, but $(5, 6, 2, 4, 1, 3, 7)$ is not.

Let $p > 2$ be a fixed prime number. Find all functions $f: \mathbb Z \to \mathbb Z_p$, where the $\mathbb Z_p$ denotes the set $\{0, 1, \ldots , p-1\}$, such that $p$ divides $f(f(n))- f(n+1) + 1$ and $f(n+p) = f(n)$ for all integers $n$. [i]Proposed by Grant Yu[/i]

Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $${10a+b:1\leq a\leq 5, 1\leq b\leq 5}$$ where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$?

Given are 100 different positive integers. We call a pair of numbers [i]good[/i] if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.) [i]Proposed by Alexander S. Golovanov, Russia[/i]