Found problems: 1800
2010 Iran MO (3rd Round), 1
suppose that $\mathcal F\subseteq X^{(k)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B,C$, at most one of $A\cap B$,$B\cap C$ and $C\cap A$ is $\phi$. for $k\le \frac{n}{2}$ prove that:
a) $|\mathcal F|\le max(1,4-\frac{n}{k})\times \dbinom{n-1}{k-1}$.(15 points)
b) find all cases of equality in a) for $k\le \frac{n}{3}$.(5 points)
2010 ELMO Shortlist, 2
For a positive integer $n$, let $s(n)$ be the number of ways that $n$ can be written as the sum of strictly increasing perfect $2010^{\text{th}}$ powers. For instance, $s(2) = 0$ and $s(1^{2010} + 2^{2010}) = 1$. Show that for every real number $x$, there exists an integer $N$ such that for all $n > N$,
\[\frac{\max_{1 \leq i \leq n} s(i)}{n} > x.\]
[i]Alex Zhu.[/i]
2007 Indonesia MO, 5
Let $ r$, $ s$ be two positive integers and $ P$ a 'chessboard' with $ r$ rows and $ s$ columns. Let $ M$ denote the maximum value of rooks placed on $ P$ such that no two of them attack each other.
(a) Determine $ M$.
(b) How many ways to place $ M$ rooks on $ P$ such that no two of them attack each other?
[Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]
2013 National Olympiad First Round, 20
The numbers $1,2,\dots, 2013$ are written on $2013$ stones weighing $1,2,\dots, 2013$ grams such that each number is used exactly once. We have a two-pan balance that shows the difference between the weights at the left and the right pans. No matter how the numbers are written, if it is possible to determine in $k$ weighings whether the weight of each stone is equal to the number that is written on the stone, what is the least possible value of $k$?
$
\textbf{(A)}\ 15
\qquad\textbf{(B)}\ 12
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ \text{None of above}
$
2003 Romania Team Selection Test, 9
Let $n\geq 3$ be a positive integer. Inside a $n\times n$ array there are placed $n^2$ positive numbers with sum $n^3$. Prove that we can find a square $2\times 2$ of 4 elements of the array, having the sides parallel with the sides of the array, and for which the sum of the elements in the square is greater than $3n$.
[i]Radu Gologan[/i]
2013 Bosnia Herzegovina Team Selection Test, 3
Prove that in the set consisting of $\binom{2n}{n}$ people we can find a group of $n+1$ people in which everyone knows everyone or noone knows noone.
1996 Iran MO (2nd round), 4
Let $n$ blue points $A_i$ and $n$ red points $B_i \ (i = 1, 2, \ldots , n)$ be situated on a line. Prove that
\[\sum_{i,j} A_i B_j \geq \sum_{i<j} A_iA_j + \sum_{i<j} B_iB_j.\]
1987 IMO Longlists, 2
Suppose we have a pack of $2n$ cards, in the order $1, 2, . . . , 2n$. A perfect shuffle of these cards changes the order to $n+1, 1, n+2, 2, . . ., n- 1, 2n, n$ ; i.e., the cards originally in the first $n$ positions have been moved to the places $2, 4, . . . , 2n$, while the remaining $n$ cards, in their original order, fill the odd positions $1, 3, . . . , 2n - 1.$
Suppose we start with the cards in the above order $1, 2, . . . , 2n$ and then successively apply perfect shuffles.
What conditions on the number $n$ are necessary for the cards eventually to return to their original order? Justify your answer.
[hide="Remark"]
Remark. This problem is trivial. Alternatively, it may be required to find the least number of shuffles after which the cards will return to the original order.[/hide]
2010 Romania National Olympiad, 1
Let $S$ be a subset with $673$ elements of the set $\{1,2,\ldots ,2010\}$. Prove that one can find two distinct elements of $S$, say $a$ and $b$, such that $6$ divides $a+b$.
2010 ELMO Problems, 3
Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence.
[list]
[*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence.
[*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list]
[i]Mitchell Lee and Benjamin Gunby.[/i]
2010 Slovenia National Olympiad, 5
Let $ABC$ be an equilateral triangle with the side of $20$ units. Amir divides this triangle into $400$ smaller equilateral triangles with the sides of $1$ unit. Reza then picks $4$ of the vertices of these smaller triangles. The vertices lie inside the triangle $ABC$ and form a parallelogram with sides parallel to the sides of the triangle $ABC.$ There are exactly $46$ smaller triangles that have at least one point in common with the sides of this parallelogram. Find all possible values for the area of this parallelogram.
[asy]
unitsize(150);
defaultpen(linewidth(0.7));
int n = 20; /* # of vertical lines, including BC */
pair A = (0,0), B = dir(-30), C = dir(30);
draw(A--B--C--cycle,linewidth(1)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0));
label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE);
for(int i = 1; i < n; ++i) {
draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n);
draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n);
draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n);
}[/asy]
[Thanks azjps for drawing the diagram.]
[hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]
IMSC 2024, 3
Alice and Bob play the following game on a square grid with $2024 \times 2024$ unit squares.
They take turns covering unit squares with stickers including their names. Alice plays the odd-numbered turns, and Bob plays the even-numbered turns. \\
On the $k$-th turn, let $n_k$ be the least integer such that $n_k\geqslant\tfrac{k}{2024}$. If there is at least one square without a sticker, then the player taking the turn:
[list = i]
[*] selects at most $n_k$ unit squares on the grid such that at least one of the chosen unit squares does not have a sticker.
[*] covers each of the selected unit squares with a sticker that has their name on it. If a selected square already has a sticker on it, then that sticker is removed first.
[/list]
At the end of their turn, a player wins if there exist $123$ unit squares containing stickers with that player's name that are placed on horizontally, vertically, or diagonally consecutive unit squares. We consider the game to be a draw if all of the unit squares are covered but no player has won yet. \\
Does Alice have a winning strategy?
[i]Proposed by Erik Paemurru, Estonia[/i]
2004 Tuymaada Olympiad, 3
Zeroes and ones are arranged in all the squares of $n\times n$ table.
All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form
[asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy]
(consisting of a square and its neighbours from left and from below)
is even.
Prove that no two rows of the table are identical.
[i]Proposed by O. Vanyushina[/i]
2011 Brazil Team Selection Test, 2
Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$.
2005 CHKMO, 2
In a school there $b$ teachers and $c$ students. Suppose that
a) each teacher teaches exactly $k$ students, and
b)for any two (distinct) students , exactly $h$ teachers teach both of them.
Prove that $\frac{b}{h}=\frac{c(c-1)}{k(k-1)}$.
2009 Romania Team Selection Test, 3
Given two integers $n\geq 1$ and $q\geq 2$, let $A=\{(a_1,\ldots ,a_n):a_i\in\{0,\ldots ,q-1\}, i=1,\ldots ,n\}$. If $a=(a_1,\ldots ,a_n)$ and $b=(b_1,\ldots ,b_n)$ are two elements of $A$, let $\delta(a,b)=\#\{i:a_i\neq b_i\}$. Let further $t$ be a non-negative integer and $B$ a non-empty subset of $A$ such that $\delta(a,b)\geq 2t+1$, whenever $a$ and $b$ are distinct elements of $B$. Prove that the two statements below are equivalent:
a) For any $a\in A$, there is a unique $b\in B$, such that $\delta (a,b)\leq t$;
b) $\displaystyle|B|\cdot \sum_{k=0}^t \binom{n}{k}(q-1)^k=q^n$
2016 Middle European Mathematical Olympiad, 4
An exam was taken by some students. Each problem was worth $1$ point for the correct answer, and $0$ points for an incorrect one.
For each question, at least one student answered it correctly. Also, there are two students with different scores on the exam.
Prove that there exists a question for which the following holds:
The average score of the students who answered the question correctly is greater than the average score of the students who didn't.
2006 Iran Team Selection Test, 6
Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex).
Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.
2006 Iran MO (3rd Round), 2
Let $B$ be a subset of $\mathbb{Z}_{3}^{n}$ with the property that for every two distinct members $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ of $B$ there exist $1\leq i\leq n$ such that $a_{i}\equiv{b_{i}+1}\pmod{3}$. Prove that $|B| \leq 2^{n}$.
2013 Tuymaada Olympiad, 8
Cards numbered from 1 to $2^n$ are distributed among $k$ children, $1\leq k\leq 2^n$, so that each child gets at least one card. Prove that the number of ways to do that is divisible by $2^{k-1}$ but not by $2^k$.
[i] M. Ivanov [/i]
1998 Irish Math Olympiad, 3
$ (a)$ Prove that $ \mathbb{N}$ can be partitioned into three (mutually disjoint) sets such that, if $ m,n \in \mathbb{N}$ and $ |m\minus{}n|$ is $ 2$ or $ 5$, then $ m$ and $ n$ are in different sets.
$ (b)$ Prove that $ \mathbb{N}$ can be partitioned into four sets such that, if $ m,n \in \mathbb{N}$ and $ |m\minus{}n|$ is $ 2,3,$ or $ 5$, then $ m$ and $ n$ are in different sets. Show, however, that $ \mathbb{N}$ cannot be partitioned into three sets with this property.
2024 Bundeswettbewerb Mathematik, 1
Arthur and Renate play a game on a $7 \times 7$ board. Arthur has two red tiles, initially placed on the cells in the bottom left and the upper right corner. Renate has two black tiles, initially placed on the cells in the bottom right and the upper left corner. In a move, a player can choose one of his two tiles and move them to a horizontally or vertically adjacent cell. The players alternate, with Arthur beginning. Arthur wins when both of his tiles are in horizontally or vertically adjacent cells after some number of moves. Can Renate prevent him from winning?
2006 Canada National Olympiad, 1
Let $ f(n,k)$ be the number of ways of distributing $ k$ candies to $ n$ children so that each child receives at most $ 2$ candies. For example $ f(3,7) \equal{} 0,f(3,6) \equal{} 1,f(3,4) \equal{} 6$. Determine the value of $ f(2006,1) \plus{} f(2006,4) \plus{} \ldots \plus{} f(2006,1000) \plus{} f(2006,1003) \plus{} \ldots \plus{} f(2006,4012)$.
2006 Mediterranean Mathematics Olympiad, 1
Every point of a plane is colored red or blue, not all with the same color.
Can this be done in such a way that, on every circumference of radius 1,
(a) there is exactly one blue point;
(b) there are exactly two blue points?
2004 Bulgaria Team Selection Test, 3
A table with $m$ rows and $n$ columns is given. At any move one chooses some empty cells such that any two of them lie in different rows and columns, puts a white piece in any of those cells and then puts a black piece in the cells whose rows and columns contain white pieces. The game is over if it is not possible to make a move. Find the maximum possible number of white pieces that can be put on the table.