Found problems: 1800
2014 JBMO TST - Turkey, 2
$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.
2010 Baltic Way, 8
In a club with $30$ members, every member initially had a hat. One day each member sent his hat to a different member (a member could have received more than one hat). Prove that there exists a group of $10$ members such that no one in the group has received a hat from another one in the group.
2012 All-Russian Olympiad, 1
$101$ wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions.
2005 Taiwan TST Round 1, 3
$n$ teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all $n$ for which this is possible.
2008 Iran MO (3rd Round), 7
A graph is called a [i]self-intersecting [/i]graph if it is isomorphic to a graph whose every edge is a segment and every two edges intersect. Notice that no edge contains a vertex except its two endings.
a) Find all $ n$'s for which the cycle of length $ n$ is self-intersecting.
b) Prove that in a self-intersecting graph $ |E(G)|\leq|V(G)|$.
c) Find all self-intersecting graphs.
[img]http://i35.tinypic.com/x43s5u.png[/img]
2008 Romania Team Selection Test, 4
Let $ G$ be a connected graph with $ n$ vertices and $ m$ edges such that each edge is contained in at least one triangle. Find the minimum value of $ m$.
2002 Irish Math Olympiad, 1
A $ 3 \times n$ grid is filled as follows. The first row consists of the numbers from $ 1$ to $ n$ arranged in ascending order. The second row is a cyclic shift of the top row: $ i,i\plus{}1,...,n,1,2,...,i\minus{}1$ for some $ i$. The third row has the numbers $ 1$ to $ n$ in some order so that in each of the $ n$ columns, the sum of the three numbers is the same. For which values of $ n$ is it possible to fill the grid in this way? For all such $ n$, determine the number of different ways of filling the grid.
1990 IMO Longlists, 89
Let $n$ be a positive integer. $S_1, S_2, \ldots, S_n$ are pairwise non-intersecting sets, and $S_k $ has exactly $k$ elements $(k = 1, 2, \ldots, n)$. Define $S = S_1\cup S_2\cup\cdots \cup S_n$. The function $f: S \to S $ maps all elements in $S_k$ to a fixed element of $S_k$, $k = 1, 2, \ldots, n$. Find the number of functions $g: S \to S$ satisfying $f(g(f(x))) = f(x).$
2002 China National Olympiad, 3
In a competition there are $18$ teams and in each round $18$ teams are divided into $9$ pairs where the $9$ matches are played coincidentally. There are $17$ rounds, so that each pair of teams play each other exactly once. After $n$ rounds, there always exists $4$ teams such that there was exactly one match played between these teams in those $n$ rounds. Find the maximum value of $n$.
2007 Indonesia TST, 3
On each vertex of a regular $ n\minus{}$gon there was a crow. Call this as initial configuration. At a signal, they all flew by and after a while, those $ n$ crows came back to the $ n\minus{}$gon, one crow for each vertex. Call this as final configuration. Determine all $ n$ such that: there are always three crows such that the triangle they formed in the initial configuration and the triangle they formed in the final configuration are both right-angled triangle.
2013 Tuymaada Olympiad, 3
The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours).
Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected.
[i]V. Dolnikov[/i]
[b]EDIT.[/b] It is confirmed by the official solution that the graph is tacitly assumed to be [b]finite[/b].
1980 IMO, 6
Given the polygons $P$ and $Q$ as shown in the grid below, cut $P$ into two polygons $P_1$ and $P_2$ such that, when pasted together differently, they form $Q$.
[asy]
import graph; size(16cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.05,xmax=15.10,ymin=-1.87,ymax=9.74;
pen cqcqcq=rgb(0.75,0.75,0.75), zzttqq=rgb(0.6,0.2,0);
draw((7,5)--(12,5)--(12,2)--(7,2)--cycle,zzttqq); draw((2,2)--(2,5)--(3,6)--(6,6)--(6,3)--(5,2)--cycle,zzttqq);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);
draw((0,8)--(0,0)); draw((0,0)--(13,0)); draw((13,0)--(13,8)); draw((13,8)--(0,8)); draw((7,5)--(12,5),zzttqq); draw((12,5)--(12,2),zzttqq); draw((12,2)--(7,2),zzttqq); draw((7,2)--(7,5),zzttqq); draw((2,2)--(2,5),zzttqq); draw((2,5)--(3,6),zzttqq); draw((3,6)--(6,6),zzttqq); draw((6,6)--(6,3),zzttqq); draw((6,3)--(5,2),zzttqq); draw((5,2)--(2,2),zzttqq);
dot((0,0),linewidth(1pt)+ds); dot((13,0),linewidth(1pt)+ds); dot((0,8),linewidth(1pt)+ds); dot((2,2),linewidth(1pt)+ds); dot((6,6),linewidth(1pt)+ds); dot((13,8),linewidth(1pt)+ds); dot((7,2),linewidth(1pt)+ds); dot((7,5),linewidth(1pt)+ds); dot((12,2),linewidth(1pt)+ds); dot((12,5),linewidth(1pt)+ds); label("$Q$",(8.42,2.56),NE*lsf,zzttqq); dot((5,2),linewidth(1pt)+ds); dot((6,3),linewidth(1pt)+ds); dot((2,5),linewidth(1pt)+ds); dot((3,6),linewidth(1pt)+ds); label("$P$",(4.65,2.74),NE*lsf,zzttqq);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]
2012 Romania Team Selection Test, 3
Let $A$ and $B$ be finite sets of real numbers and let $x$ be an element of $A+B$. Prove that \[|A\cap (x-B)|\leq \frac{|A-B|^2}{|A+B|}\] where $A+B=\{a+b: a\in A, b\in B\}$, $x-B=\{x-b: b\in B\}$ and $A-B=\{a-b: a\in A, b\in B\}$.
2014 Moldova Team Selection Test, 4
On a circle $n \geq 1$ real numbers are written, their sum is $n-1$. Prove that one can denote these numbers as $x_1, x_2, ..., x_n$ consecutively, starting from a number and moving clockwise, such that for any $k$ ($1\leq k \leq n$) $ x_1 + x_2+...+x_k \geq k-1$.
2007 Singapore Team Selection Test, 3
Let $ a_1, a_2,\ldots ,a_8$ be $8$ distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every $4$ of the $8$ points form a quadrilateral which is called a [i]quad[/i]. If two chords, each joining a pair of the $8$ points, intersect, the point of intersection is called a [i]bullet[/i]. Suppose some of the bullets are coloured red. For each pair $(i j)$, with $ 1 \le i < j \le 8$, let $r(i,j)$ be the number of quads, each containing $ a_i, a_j$ as vertices, whose diagonals intersect at a red bullet. Determine the smallest positive integer $n$ such that it is possible to colour $n$ of the bullets red so that $r(i,j)$ is a constant for all pairs $(i,j)$.
2000 Mediterranean Mathematics Olympiad, 1
Let $F=\{1,2,...,100\}$ and let $G$ be any $10$-element subset of $F$. Prove that there exist two disjoint nonempty subsets $S$ and $T$ of $G$ with the same sum of elements.
2005 Mediterranean Mathematics Olympiad, 3
Let $A_1,A_2,\ldots , A_n$ $(n\geq 3)$ be finite sets of positive integers. Prove, that
\[ \displaystyle \frac{1}{n} \left( \sum_{i=1}^n |A_i|\right) + \frac{1}{{{n}\choose{3}}}\sum_{1\leq i < j < k \leq n} |A_i \cap A_j \cap A_k| \geq \frac{2}{{{n}\choose{2}}}\sum_{1\leq i < j \leq n}|A_i \cap A_j| \]
holds, where $|E|$ is the cardinality of the set $E$
2012 IberoAmerican, 3
Let $n$ to be a positive integer. Given a set $\{ a_1, a_2, \ldots, a_n \} $ of integers, where $a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \},$ $\forall i$, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to $0$. If all of these sums have different remainders when divided by $2^n$, we say that $\{ a_1, a_2, \ldots, a_n \} $ is [i]$n$-complete[/i].
For each $n$, find the number of [i]$n$-complete[/i] sets.
2012 Iran MO (3rd Round), 3
Prove that if $n$ is large enough, then for each coloring of the subsets of the set $\{1,2,...,n\}$ with $1391$ colors, two non-empty disjoint subsets $A$ and $B$ exist such that $A$, $B$ and $A\cup B$ are of the same color.
2004 Czech and Slovak Olympiad III A, 3
Given a circle $S$ and its $121$ chords $P_i (i=1,2,\ldots,121)$, each with a point $A_i(i=1,2,\ldots,121)$ on it. Prove that there exists a point $X$ on the circumference of $S$ such that: there exist $29$ distinct indices $1\le k_1\le k_2\le\ldots\le k_{29}\le 121$, such that the angle formed by ${A_{k_j}}X$ and ${P_{k_j}}$ is smaller than $21$ degrees for every $j=1,2,\ldots,29$.
2001 Iran MO (2nd round), 3
Suppose a table with one row and infinite columns. We call each $1\times1$ square a [i]room[/i]. Let the table be finite from left. We number the rooms from left to $\infty$. We have put in some rooms some coins (A room can have more than one coin.). We can do $2$ below operations:
$a)$ If in $2$ adjacent rooms, there are some coins, we can move one coin from the left room $2$ rooms to right and delete one room from the right room.
$b)$ If a room whose number is $3$ or more has more than $1$ coin, we can move one of its coins $1$ room to right and move one other coin $2$ rooms to left.
$i)$ Prove that for any initial configuration of the coins, after a finite number of movements, we cannot do anything more.
$ii)$ Suppose that there is exactly one coin in each room from $1$ to $n$. Prove that by doing the allowed operations, we cannot put any coins in the room $n+2$ or the righter rooms.
2007 Croatia Team Selection Test, 6
$\displaystyle 2n$ students $\displaystyle (n \geq 5)$ participated at table tennis contest, which took $\displaystyle 4$ days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this:
- there is only one winner;
- there are $\displaystyle 3$ students on the second place;
- no student lost all $\displaystyle 4$ matches.
How many students won only a single match and how many won exactly $\displaystyle 2$ matches? (In the above conditions)
2011 Argentina Team Selection Test, 2
A wizard kidnaps $31$ members from party $A$, $28$ members from party $B$, $23$ members from party $C$, and $19$ members from party $D$, keeping them isolated in individual rooms in his castle, where he forces them to work.
Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved).
a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong.
b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.
2009 Czech and Slovak Olympiad III A, 5
At every vertex $A_k(1\le k\le n)$ of a regular $n$-gon, $k$ coins are placed. We can do the following operation: in each step, one can choose two arbitrarily coins and move them to their adjacent vertices respectively, one clockwise and one anticlockwise. Find all positive integers $n$ such that after a finite number of operations, we can reach the following configuration: there are $n+1-k$ coins at vertex $A_k$ for all $1\le k\le n$.
2009 Indonesia TST, 2
Prove that there exists two different permutations $ (a_1,a_2,\dots,a_{2009})$ and $ (b_1,b_2,\dots,b_{2009})$ of $ (1,2,\dots,2009)$ such that \[ \sum_{i\equal{}1}^{2009}i^i a_i \minus{} \sum_{i\equal{}1}^{2009} i^i b_i\] is divisible by $ 2009!$.