$60$ children participate in a summer camp. Among any $10$ of the children there are three or more who live in the same block. Prove that there must be $15$ or more children from the same block. (Folklore)

Let $n$ be a natural number greater than 6. $X$ is a set such that $|X| = n$. $A_1, A_2, \ldots, A_m$ are distinct 5-element subsets of $X$. If $m > \frac{n(n - 1)(n - 2)(n - 3)(4n - 15)}{600}$, prove that there exists $A_{i_1}, A_{i_2}, \ldots, A_{i_6}$ $(1 \leq i_1 < i_2 < \cdots, i_6 \leq m)$, such that $\bigcup_{k = 1}^6 A_{i_k} = 6$.

There are $100$ lightbulbs $B_1,\ldots, B_{100}$ spaced evenly around a circle in this order. Additionally, there are $100$ switches $S_1,\ldots, S_{100}$ such that for all $1\leq i\leq 100$, switch $S_i$ toggles the states of lights $B_{i-1}$ and $B_{i+1}$ (where here $B_{101} = B_1$). Suppose David chooses whether to flick each switch with probability $\tfrac12$. What is the expected number of lightbulbs which are on at the end of this process given that not all lightbulbs are off?

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]

The Banana Republic language has more words than letters in its alphabet. Prove that there is a natural number $k$ for which we can choose $k$ different words that use exactly $k$ different letters.

Luke chose a set of three different dates $a,b,c$ in the month of May, where in any year, if one makes a calendar with a sheet of grid paper the centers of the cells with dates $a,b,c$ would form an isosceles right triangle or a straight line. How many sets can be chosen? [img]https://cdn.artofproblemsolving.com/attachments/7/3/dbf90fdc81fc0f0d14c32020b69df53b67b397.png[/img]

Given a set of $n$ positive numbers. For each its nonempty subset consider the sum of all the subset's numbers. Prove that you can divide those sums onto $n$ groups in such a way, that the least sum in every group is not less than a half of the greatest sum in the same group.

You have $100$ equal rods. It is allowed to split each rod into two or three shorter rods, not necessarily the same. The objective is that by rearranging the pieces (and using them all) $q>200$ can be assembled new rods, all of equal length. Find the values ​​of $q$ for whom this can be done.

Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]

The nonnegative integer $n$ and $ (2n + 1) \times (2n + 1)$ chessboard with squares colored alternatively black and white are given. For every natural number $m$ with $1 < m < 2n+1$, an $m \times m$ square of the given chessboard that has more than half of its area colored in black, is called a $B$-square. If the given chessboard is a $B$-square, find in terms of $n$ the total number of $B$-squares of this chessboard.

Seven equally-spaced points are drawn on a circle of radius $1$. Three distinct points are chosen uniformly at random. What is the probability that the center of the circle lies in the triangle formed by the three points?

Given $r\in\mathbb{R}$. Alice and Bob plays the following game: An equation with blank is written on the blackboard as below: $$S=|\Box-\Box|+|\Box-\Box|+|\Box-\Box|$$ Every round, Alice choose a real number from $[0,1]$ (not necessary to be different from the numbers chosen before) and Bob fill it in an empty box. After 6 rounds, every blank is filled and $S$ is determined at the same time. If $S\ge r$ then Alice wins, otherwise Bob wins. Find all $r$ such that Alice can guarantee her victory.

Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop." [list][b](a)[/b] Can Andy guarantee that after he says "stop," no card is in its initial spot? [b](b)[/b] Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to the empty spot?[/list]

There is a football championship with $6$ teams involved, such that for any $2$ teams $A$ and $B$, $A$ plays with $B$ and $B$ plays with $A$ ($2$ such games are distinct). After every match, the winning teams gains $3$ points, the loosing team gains $0$ points and if there is a draw, both teams gain $1$ point each.\\ \\ In the end, the team standing on the last place has $12$ points and there are no $2$ teams that scored the same amount of points.\\ \\ For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations. $\textit{(Andrei Bâra)}$

Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

A magician and his assistent are performing the following trick.There is a row of 12 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistent knows which boxes contain coins. The magician returns, and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects 4 boxes, which are simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always succesfully perform the trick.

Let $AB$ be a segment of length $1$. Several particles start moving simultaneously at constant speeds from $A$ up to$ B$. As soon as a particle reaches $B$ , turns around and goes to $A$; when it reaches $A$, begins to move again towards $ B$ , and so on indefinitely. Find all rational numbers$ r>1$ such that there exists an instant $t$ with the following property: For each $n\ge 1$ , if $n+1$ particles with constant speeds $1,r,r^2,…,r^n$ move as described, at instant $t$, they all lie at the same interior point of segment $AB$.

On the board we write a series of $n$ numbers, where $n \geq 40$, and each one of them is equal to either $1$ or $-1$, such that the following conditions both hold: (i) The sum of every $40$ consecutive numbers is equal to $0$. (ii) The sum of every $42$ consecutive numbers is not equal to $0$. We denote by $S_n$ the sum of the $n$ numbers of the board. Find the maximum possible value of $S_n$ for all possible values of $n$.

Let $n$ be a natural number. We have $n$ colors. Each of the numbers $1, 2, 3,... , 1000$ was colored with one of the $n$ colors. It is known that, for any two distinct numbers, if one divides the other then these two numbers have different colors. Determine the smallest possible value of $n$.

[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square. [b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$. [b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img] [b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Starting at a vertex $x_0$, we walk over the edges of a connected graph $G$ according to the following rules: 1. We never walk the same edge twice in the same direction. 2. Once we reach a vertex $x \ne x_0$, never visited before, we mark the edge by which we come to $x$. We can use this marked edge to leave vertex $x$ only if we already have traversed, in both directions, all other edges incident to $x$. Show that, regardless of the path followed, we will always be stuck at $x_0$ and that all other edges will have been traveled in both directions.

In how many ways can a set of $12$ elements be partitioned into six two-element subsets?

We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$. With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elements. These elements that we removed are the elements of $M_1$. In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define $M_2$. We continue similarly until there are no more elements in $A_1,A_2,...,A_{160}$, thus defining the sets $M_1,M_2,...,M_n$. Find the minimum value of $n$.

Sixteen squares of an $8\times 8$ chessboard are chosen so that there are exactly lwo in each row and two in each column. Prove that eight white pawns and eight black pawns can be placed on these sixteen squares so that there is one white pawn and one black pawn in each row and in cach colunm.

A rectangle of dimensions $2024 \times 2023$ is filled with pieces of the following types: [asy] size(200); // Figure (A) draw((0,0)--(4,0)--(4,1)--(0,1)--cycle); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((3,0)--(3,1)); // Figure (B) draw((6,0)--(8,0)--(8,2)--(6,2)--cycle); draw((7,0)--(7,2)); draw((6,1)--(8,1)); // Figure (C) draw((10,0)--(12,0)--(12,1)--(11,1)--(11,2)--(9,2)--(9,1)--(10,1)--cycle); draw((10,0)--(10,1)); draw((11,0)--(11,1)); draw((10,1)--(11,1)); draw((9,1)--(9,2)); draw((10,1)--(10,2)); draw((11,0)--(12,0)); draw((10,1)--(12,1)); // Labeling label("(A)", (2, -0.5)); label("(B)", (7, -0.5)); label("(C)", (10.5, -0.5)); [/asy] Each piece can be turned arround, and each square has side length $1$. Is it possible to use exactly 2023 pieces of type $(A)$?