Found problems: 1800
2008 CentroAmerican, 3
There are 2008 bags numbered from 1 to 2008, with 2008 frogs in each one of them. Two people play in turns. A play consists in selecting a bag and taking out of it any number of frongs (at least one), leaving $ x$ frogs in it ($ x\geq 0$). After each play, from each bag with a number higher than the selected one and having more than $ x$ frogs, some frogs scape until there are $ x$ frogs in the bag. The player that takes out the last frog from bag number 1 looses. Find and explain a winning strategy.
2008 Mongolia Team Selection Test, 2
Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.
2023 German National Olympiad, 3
For a competition a school wants to nominate a team of $k$ students, where $k$ is a given positive integer. Each member of the team has to compete in the three disciplines juggling, singing and mental arithmetic. To qualify for the team, the $n \ge 2$ students of the school compete in qualifying competitions, determining a unique ranking in each of the three disciplines. The school now wants to nominate a team satisfying the following condition:
$(*)$ [i]If a student $X$ is not nominated for the team, there is a student $Y$ on the team who defeated $X$ in at least two disciplines.[/i]
Determine all positive integers $n \ge 2$ such that for any combination of rankings, a team can be chosen to satisfy the condition $(*)$, when
a) $k=2$,
b) $k=3$.
2002 Tournament Of Towns, 1
All the species of plants existing in Russia are catalogued (numbered by integers from $2$ to $2000$ ; one after another, without omissions or repetitions). For any pair of species the gcd of their catalogue numbers was calculated and recorded but the catalogue numbers themselves were lost. Is it possible to restore the catalogue numbers from the data in hand?
2017 Baltic Way, 6
Fifteen stones are placed on a $4 \times 4$ board, one in each cell, the remaining cell being empty. Whenever two stones are on neighbouring cells (having a common side), one may jump over the other to the opposite neighbouring cell, provided this cell is empty. The stone jumped over is removed from the board.
For which initial positions of the empty cell is it possible to end up with exactly one stone on the board?
2003 Kazakhstan National Olympiad, 3
Two square sheets have areas equal to $ 2003$. Each of the sheets is arbitrarily divided into $ 2003$ nonoverlapping polygons, besides, each of the polygons has an unitary area. Afterward, one overlays two sheets, and it is asked to prove that the obtained double layer can be punctured $ 2003$ times, so that each of the $ 4006$ polygons gets punctured precisely once.
2012 JBMO TST - Turkey, 2
Let $S=\{1,2,3,\ldots,2012\}.$ We want to partition $S$ into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of $2.$ Find the number of such partitions.
1976 Canada National Olympiad, 3
Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and each grade eight student scored the same number of points as his classmates. How many students for grade eight participated in the chess tournament? Is the solution unique?
2011 All-Russian Olympiad, 4
Ten cars are moving at the road. There are some cities at the road. Each car is moving with some constant speed through cities and with some different constant speed outside the cities (different cars may move with different speed). There are 2011 points at the road. Cars don't overtake at the points. Prove that there are 2 points such that cars pass through these points in the same order.
[i]S. Berlov[/i]
2007 QEDMO 4th, 6
Any two islands of the Chaos Archipelago are connected by a bridge - a red bridge or a blue bridge. Show that at least one of the following two assertions holds:
$\mathcal{A}_{1}$: For any two islands $a$ and $b$, we can reach $b$ from $a$ through at most $3$ red bridges (and no blue bridges).
$\mathcal{A}_{2}$: For any two islands $a$ and $b$, we can reach $b$ from $a$ through at most $2$ blue bridges (and no red bridges).
[i]Alternative formulation:[/i] Let $G$ be a graph. Prove that the diameter of $G$ is $\leq 3$ or the diameter of the complement of $G$ is $\leq 2$.
[i]Note.[/i] This problem is the main Theorem in
Frank Harary, Robert W. Robinson, [i]The Diameter of a Graph and its Complement[/i], The American Mathematical Monthly, Vol. 92, No. 3. (Mar., 1985), pp. 211-212.
darij
1998 Korea - Final Round, 3
Let $F_n$ be the set of all bijective functions $f:\left\{1,2,\ldots,n\right\}\rightarrow\left\{1,2,\ldots,n\right\}$ that satisfy the conditions:
1. $f(k)\leq k+1$ for all $k\in\left\{1,2,\ldots,n\right\}$
2. $f(k)\neq k$ for all $k\in\left\{2,3,\ldots,n\right\}$
Find the probability that $f(1)\neq1$ for $f$ randomly chosen from $F_n$.
2007 Romania National Olympiad, 4
a) For a finite set of natural numbers $S$, denote by $S+S$ the set of numbers $z=x+y$, where $x,y\in S$. Let $m=|S|$. Show that $|S+S|\leq \frac{m(m+1)}{2}$.
b) Let $m$ be a fixed positive integer. Denote by $C(m)$ the greatest integer $k\geq 1$ for which there exists a set $S$ of $m$ integers, such that $\{1,2,\ldots,k\}\subseteq S\cup(S+S)$. For example, $C(3)=8$, with $S=\{1,3,4\}$. Show that $\frac{m(m+6)}{4}\leq C(m) \leq \frac{m(m+3)}{2}$.
2008 Pan African, 3
Let $a,b,c$ be three positive integers such that $a<b<c$. Consider the the sets $A,B,C$ and $X$, defined as follows: $A=\{ 1,2,\ldots ,a \}$, $B=\{a+1,a+2,\ldots,b\}$, $C=\{b+1,b+2,\ldots ,c\}$ and $X=A\cup B\cup C$.
Determine, in terms of $a,b$ and $c$, the number of ways of placing the elements of $X$ in three boxes such that there are $x,y$ and $z$ elements in the first, second and third box respectively, knowing that:
i) $x\le y\le z$;
ii) elements of $B$ cannot be put in the first box;
iii) elements of $C$ cannot be put in the third box.
2007 Czech-Polish-Slovak Match, 5
For which $n\in\{3900, 3901,\cdots, 3909\}$ can the set $\{1, 2, . . . , n\}$ be partitioned into (disjoint) triples in such a way that in each triple one of the numbers equals the sum of the other two?
1986 IMO Longlists, 72
A one-person game with two possible outcomes is played as follows:
After each play, the player receives either $a$ or $b$ points, where $a$ and $b$ are integers with $0 < b < a < 1986$. The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive plays. It is observed that every integer $x \geq 1986$ can be obtained as the total score whereas $1985$ and $663$ cannot. Determine $a$ and $b.$
2011 ELMO Shortlist, 3
Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows.
(Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.)
[i]Linus Hamilton.[/i]
2011 Baltic Way, 9
Given a rectangular grid, split into $m\times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions:
[list]
[*]All squares touching the border of the grid are coloured black.
[*]No four squares forming a $2\times 2$ square are coloured in the same colour.
[*]No four squares forming a $2\times 2$ square are coloured in such a way that only diagonally touching
squares have the same colour.[/list]
Which grid sizes $m\times n$ (with $m,n\ge 3$) have a valid colouring?
2010 All-Russian Olympiad, 2
There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.
2007 Italy TST, 2
In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C }$ if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$.
(a) Find the minimum of cyclical 3-subset (depending on $n$);
(b) Find the maximum of cyclical 3-subset (depending on $n$).
1999 Baltic Way, 8
We are given $1999$ coins. No two coins have the same weight. A machine is provided which allows us with one operation to determine, for any three coins, which one has the middle weight. Prove that the coin that is the $1000$th by weight can be determined using no more than $1000000$ operations and that this is the only coin whose position by weight can be determined using this machine.
2007 All-Russian Olympiad Regional Round, 8.4
On the chessboard, $ 32$ black pawns and $ 32$ white pawns are arranged. In every move, a pawn can capture another pawn of the opposite color, moving diagonally to an adjacent square where the captured one stands. White pawns move only in upper-left or upper-right directions, while black ones can move in down-left or in down-right directions only; the captured pawn is removed from the board. A pawn cannot move without capturing an opposite pawn. Find the least possible number of pawns which can stay on the chessboard.
2013 Olympic Revenge, 1
Let $n$ to be a positive integer. A family $\wp$ of intervals $[i, j]$ with $0 \le i < j \le n$ and $i$, $j$ integers is considered [i]happy[/i] if, for any $I_1 = [i_1, j_1] \in \wp$ and $I_2 = [i_2, j_2] \in \wp$ such that $I_1 \subset I_2$, we have $i_1 = i_2$ [b]or[/b] $j_1 = j_2$.
Determine the maximum number of elements of a [i]happy[/i] family.
2022 Turkey Team Selection Test, 5
On a circle, 2022 points are chosen such that distance between two adjacent points is always the same. There are $k$ arcs, each having endpoints on chosen points, with different lengths. Arcs do not contain each other. What is the maximum possible number of $k$?
2010 Iran MO (3rd Round), 5
suppose that $\mathcal F\subseteq p(X)$ and $|X|=n$. prove that if $|\mathcal F|>\sum_{i=0}^{k-1}\dbinom{n}{i}$ then there exist $Y\subseteq X$ with $|Y|=k$ such that $p(Y)=\mathcal F\cap Y$ that $\mathcal F\cap Y=\{F\cap Y:F\in \mathcal F\}$(20 points)
you can see this problem also here:
COMBINATORIAL PROBLEMS AND EXERCISES-SECOND EDITION-by LASZLO LOVASZ-AMS CHELSEA PUBLISHING- chapter 13- problem 10(c)!!!
2012 China Team Selection Test, 3
In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$.
For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.