Found problems: 14842
2021 Science ON all problems, 4
Take $k\in \mathbb{Z}_{\ge 1}$ and the sets $A_1,A_2,\dots, A_k$ consisting of $x_1,x_2,\dots ,x_k$ positive integers, respectively. For any two sets $A$ and $B$, define $A+B=\{a+b~|~a\in A,~b\in B\}$.
Find the least and greatest number of elements the set $A_1+A_2+\dots +A_k$ may have.
[i] (Andrei Bâra)[/i]
2021 Czech and Slovak Olympiad III A, 5
We call a string of characters [i]neat [/i] when it has an even length and its first half is identical to the other half (eg. [i]abab[/i]). We call a string [i]nice [/i] if it can be split on several neat strings (e.g. [i]abcabcdedef [/i]to [i]abcabc[/i], [i]dede[/i], and [i]ff[/i]). By string [i]reduction[/i] we call an operation in which we wipe two identical adjacent characters from the string (e.g. the string [i]abbac[/i] can be reduced to [i]aac[/i] and further to [i]c[/i]). Prove any string containing each of its characters in even numbers can be obtained by a series of reductions from a suitable nice string.
(Martin Melicher)
1990 ITAMO, 1
A cube of edge length $3$ consists of $27$ unit cubes. Find the number of lines passing through exactly three centers of these $27$ cubes, as well as the number of those passing through exactly two such centers.
1998 Estonia National Olympiad, 3
On a closed track, clockwise, there are five boxes $A, B, C, D$ and $E$, and the length of the track section between boxes $A$ and $B$ is $1$ km, between $B$ and $C$ - $5$ km, between $C$ and $D$ - $2$ km, between $D$ and $E$ - $10$ km, and between $E$ and $A$ - $3$ km. On the track, they drive in a clockwise direction, the race always begins and ends in the box. What box did you start from if the length of the race was exactly $1998$ km?
2018 Israel Olympic Revenge, 2
Is it possible to disassemble and reassemble a $4\times 4\times 4$ Rubik's Cuble in at least $577,800$ non-equivalent ways?
Notes:
1. When we reassemble the cube, a corner cube has to go to a corner cube, an edge cube must go to an edge cube and a central cube must go to a central cube.
2. Two positions of the cube are called equivalent if they can be obtained from one two another by rotating the faces or layers which are parallel to the faces.
2011 Tournament of Towns, 3
A balance and a set of pairwise different weights are given. It is known that for any pair of weights from this set put on the left pan of the balance, one can counterbalance them by one or several of the remaining weights put on the right pan. Find the least possible number of weights in the set.
2017 Hanoi Open Mathematics Competitions, 9
Cut off a square carton by a straight line into two pieces, then cut one of two pieces into two small pieces by a straight line, ect. By cutting $2017$ times we obtain $2018$ pieces. We write number $2$ in every triangle, number 1 in every quadrilateral, and $0$ in the polygons. Is the sum of all inserted numbers always greater than $2017$?
1992 All Soviet Union Mathematical Olympiad, 568
A cinema has its seats arranged in $n$ rows $\times m$ columns. It sold mn tickets but sold some seats more than once. The usher managed to allocate seats so that every ticket holder was in the correct row or column. Show that he could have allocated seats so that every ticket holder was in the correct row or column and at least one person was in the correct seat. What is the maximum $k$ such that he could have always put every ticket holder in the correct row or column and at least $k$ people in the correct seat?
2018 China Team Selection Test, 4
Suppose $A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \}$ and $\left | A_i \right |=2, i=1,2,\cdots ,n$, satisfying that $$A_i + A_j, \; 1 \le i \le j \le n ,$$ are distinct from each other. $A + B = \left \{ a+b|a\in A,\,b\in B \right \}$. Determine the maximal value of $n$.
OIFMAT I 2010, 7
$ 15 $ teams participate in a soccer league. Each team plays each of the remaining teams exactly once. If a team beats another team in a match they receive $ 3 $ points, while the loser receives $ 1 $ point. In the event of a tie, both teams receive $ 2 $ points. When all possible league matches are held, the following can be observed:
$\bullet$ No two teams have finished with the same amount of points.
$\bullet$ Each team finished the league with at least $ 21 $ points.
Let $W$ be the team that finished the league with the highest score. Determine how many points $W$ scored and show that there were at least four ties in the league.
1999 USAMO, 5
The Y2K Game is played on a $1 \times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.
2009 Tournament Of Towns, 3
Alex is going to make a set of cubical blocks of the same size and to write a digit on each of their faces so that it would be possible to form every $30$-digit integer with these blocks. What is the minimal number of blocks in a set with this property? (The digits $6$ and $9$ do not turn one into another.)
2002 Hong kong National Olympiad, 2
In conference there $n>2$ mathematicians. Every two mathematicians communicate in one of the $n$ offical languages of the conference. For any three different offical languages the exists three mathematicians who communicate with each other in these three languages. Find all $n$ such that this is possible.
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.
1996 Kurschak Competition, 2
Two countries ($A$ and $B$) organize a conference, and they send an equal number of participants. Some of them have known each other from a previous conference. Prove that one can choose a nonempty subset $C$ of the participants from $A$ such that one of the following holds:
[list][*]the participants from $B$ each know an even number of people in $C$,
[*]the participants from $B$ each know an odd number of participants in $C$.[/list]
2023 India IMO Training Camp, 2
In a school, every pair of students are either friends or strangers. Friendship is mutual, and no student is friends with themselves. A sequence of (not necessarily distinct) students $A_1, A_2, \dots, A_{2023}$ is called [i]mischievous[/i] if
$\bullet$ Total number of friends of $A_1$ is odd.
$\bullet$ $A_i$ and $A_{i+1}$ are friends for $i=1, 2, \dots, 2022$.
$\bullet$ Total number of friends of $A_{2023}$ is even.
Prove that the total number of [i]mischievous[/i] sequences is even.
2003 Brazil National Olympiad, 2
Let $S$ be a set with $n$ elements. Take a positive integer $k$. Let $A_1, A_2, \ldots, A_k$ be any distinct subsets of $S$. For each $i$ take $B_i = A_i$ or $B_i = S - A_i$. Find the smallest $k$ such that we can always choose $B_i$ so that $\bigcup_{i=1}^k B_i = S$, no matter what the subsets $A_i$ are.
2002 India IMO Training Camp, 18
Consider the square grid with $A=(0,0)$ and $C=(n,n)$ at its diagonal ends. Paths from $A$ to $C$ are composed of moves one unit to the right or one unit up. Let $C_n$ (n-th catalan number) be the number of paths from $A$ to $C$ which stay on or below the diagonal $AC$. Show that the number of paths from $A$ to $C$ which cross $AC$ from below at most twice is equal to $C_{n+2}-2C_{n+1}+C_n$
1998 Tournament Of Towns, 6
(a) Two people perform a card trick. The first performer takes $5$ cards from a $52$-card deck (previously shuffled by a member of the audience) , looks at them, and arranges them in a row from left to right: one face down (not necessarily the first one) , the others face up . The second performer guesses correctly the card which is face down. Prove that the performers can agree on a system which always makes this possible.
(b) For their second trick, the first performer arranges four cards in a row, face up, the fifth card is kept hidden. Can they still agree on a system which enables the second performer to correctly guess the hidden card?
(G Galperin)
2018 Iran MO (3rd Round), 3
Find the smallest positive integer $n$ such that we can write numbers $1,2,\dots ,n$ in a 18*18 board such that:
i)each number appears at least once
ii)In each row or column,there are no two numbers having difference 0 or 1
2018 PUMaC Team Round, 13
Consider a 10-dimensional \(10 \times 10 \times \cdots \times 10 \) cube consisting of \(10^{10}\) unit cubes, such that one cube \(A\) is centered at the origin, and one cube \(B\) is centered at \((9, 9, 9, 9, 9, 9, 9, 9, 9, 9)\). Paint \(A\) red and remove \(B\), leaving an empty space. Let a move consist of taking a cube adjacent to the empty space and placing it into the empty space, leaving the space originally contained by the cube empty. What is the minimum number of moves required to result in a configuration where the cube centered at \((9, 9, 9, 9, 9, 9, 9, 9, 9, 9)\) is red?
2001 Tournament Of Towns, 5
On a square board divided into $15 \times 15$ little squares there are $15$ rooks that do not attack each other. Then each rook makes one move like that of a knight. Prove that after this is done a pair of rooks will necessarily attack each other.
2017 BMT Spring, 2
Each BMT, every student chooses one of three focus rounds to take. Bob plans to attend BMT for the next $4$ years and wants to gure out what focus round to take each year. Given that he wants to take each focus round at least once, how many ways can he choose which round to take each year?
1967 IMO Shortlist, 6
On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?
1970 Dutch Mathematical Olympiad, 5
$2n$ clubs want to play a league. Each club must play every other club exactly once. Each club is only allowed to play one game per day. Prove that the competition can be completed in $2n - 1$ days.