Found problems: 1800
2012 Iran Team Selection Test, 3
We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set
\[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero).
[i]Proposed by Javad Abedi[/i]
1987 Canada National Olympiad, 4
On a large, flat field $n$ people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When $n$ is odd show that there is at least one person left dry. Is this always true when $n$ is even?
1998 Bulgaria National Olympiad, 3
The sides and diagonals of a regular $n$-gon $R$ are colored in $k$ colors so that:
(i) For each color $a$ and any two vertices $A$,$B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$.
(ii) The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors.
Prove that $k\leq 2$.
2005 Tuymaada Olympiad, 3
The organizers of a mathematical congress found that if they accomodate any participant in a room the rest can be accomodated in double rooms so that 2 persons living in each room know each other. Prove that every participant can organize a round table on graph theory for himself and an even number of other people so that each participant of the round table knows both his neigbours.
[i]Proposed by S. Berlov, S. Ivanov[/i]
2010 Iran Team Selection Test, 4
$S,T$ are two trees without vertices of degree 2. To each edge is associated a positive number which is called length of this edge. Distance between two arbitrary vertices $v,w$ in this graph is defined by sum of length of all edges in the path between $v$ and $w$. Let $f$ be a bijective function from leaves of $S$ to leaves of $T$, such that for each two leaves $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $f(u), f(v)$ in $T$. Prove that there is a bijective function $g$ from vertices of $S$ to vertices of $T$ such that for each two vertices $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $g(u)$ and $g(v)$ in $T$.
2006 Moldova Team Selection Test, 4
Let $A=\{1,2,\ldots,n\}$. Find the number of unordered triples $(X,Y,Z)$ that satisfy $X\bigcup Y \bigcup Z=A$
2004 German National Olympiad, 6
Is there a circle which passes through five points with integer co-ordinates?
2008 Stars Of Mathematics, 3
Let $ k > 1$ be an integer, and consider the in finite array given by the integer lattice in the first quadrant of the plane, filled with real numbers. The array is said to be constant if all its elements are equal in value. The array is said to be $ k$-balanced if it is non-constant, and the sums of the elements of any $ k\times k$ sub-square have a constant value $ v_k$. An array which is both $ p$-balanced and $ q$-balanced will be said to be $ (p, q)$-balanced, or just doubly-balanced, if there is no confusion as to which $ p$ and $ q$ are meant. If $p, q$ are relatively prime, the array is said to be co-prime. We will call $ (M\times N)$-seed a $ M \times N$ array, anchored with its lower left corner in the origin of the plane, which extended through periodicity in both dimensions in the plane results into a $ (p, q)$-balanced array; more precisely, if we denote the numbers in the array by $ a_{ij}$ , where $ i, j$ are the coordinates of the lower left corner of the unit square they lie in, we have, for all non-negative integers $ i, j$
\[ a_{i \plus{} M,j} \equal{} a_{i,j} \equal{} a_{i,j \plus{} N}\]
(a) Prove that $ q^2v_p \equal{} p^2v_q$ for a $ (p, q)$-balanced array.
(b) Prove that more than two different values are used in a co-prime $ (p,q)$-balanced array. Show that this is no longer true if $ (p, q) > 1$.
(c) Prove that any co-prime $ (p, q)$-balanced array originates from a seed.
(d) Show there exist $ (p, q)$-balanced arrays (using only three different values) for arbitrary values $ p, q$.
(e) Show that neither a $ k$-balanced array, nor a $ (p, q)$-balanced array if $ (p, q) > 1$, need originate from a seed.
(f) Determine the minimal possible value $ T$ for a square $ (T\times T)$-seed resulting in a co-prime $ (p, q)$-balanced array, when $p,q$ are both prime.
(g) Show that for any relatively prime $ p, q$ there must exist a co-prime $ (p, q)$-balanced array originating from a square $ (T\times T)$-seed, with no lesser $ (M\times N)$-seed available ($ M\leq T, N\leq T$ and $MN< T^2$).
[i]Dan Schwarz[/i]
2013 India IMO Training Camp, 1
Let $n \ge 2$ be an integer. There are $n$ beads numbered $1, 2, \ldots, n$. Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no flipping allowed). For example, with $n \ge 5$, the necklace with four beads $1, 5, 3, 2$ in the clockwise order is same as the one with $5, 3, 2, 1$ in the clockwise order, but is different from the one with $1, 2, 3, 5$ in the clockwise order.
We denote by $D_0(n)$ (respectively $D_1(n)$) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least $3$. Prove that $n - 1$ divides $D_1(n) - D_0(n)$.
1979 Miklós Schweitzer, 1
Let the operation $ f$ of $ k$ variables defined on the set $ \{ 1,2,\ldots,n \}$ be called $ \textit{friendly}$ toward the binary relation $ \rho$ defined on the same set if \[ f(a_1,a_2,\ldots,a_k) \;\rho\ \;f(b_1,b_2,\ldots,b_k)\] implies $ a_i \; \rho \ b_i$ for at least one $ i,1\leq i \leq k$. Show that if the operation $ f$ is friendly toward the relations "equal to" and "less than," then it is friendly toward all binary relations.
[i]B. Csakany[/i]
2010 ELMO Shortlist, 7
The game of circulate is played with a deck of $kn$ cards each with a number in $1,2,\ldots,n$ such that there are $k$ cards with each number. First, $n$ piles numbered $1,2,\ldots,n$ of $k$ cards each are dealt out face down. The player then flips over a card from pile $1$, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the $k$ cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game?
[i]Brian Hamrick.[/i]
2006 Iran Team Selection Test, 6
Let $G$ be a tournoment such that it's edges are colored either red or blue.
Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.
2015 China Team Selection Test, 6
There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given:
\\(1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$)
\\(2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$).
\\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)
2009 Olympic Revenge, 5
Thin and Fat eat a pizza of $2n$ pieces. Each piece contains a distinct amount of olives between $1$ and $2n$. Thin eats the first piece, and the two players alternately eat a piece neighbor of an eaten piece. However, neither Thin nor Fat like olives, so they will choose pieces that minimizes the total amount of olives they eat. For each arrangement $\sigma$ of the olives, let $s(\sigma)$ the minimal amount of olives that Thin can eat, considering that both play in the best way possible. Let $S(n)$ the maximum of $s(\sigma)$, considering all arrangements.
$a)$ Prove that $n^2-1+\lfloor \frac{n}{2} \rfloor \le S(n) \le n^2+\lfloor \frac{n}{2} \rfloor$
$b)$ Prove that $S(n)=n^2-1+\frac{n}{2}$ for each even n.
2012 Indonesia TST, 1
A cycling group that has $4n$ members will have several cycling events, such that:
a) Two cycling events are done every week; once on Saturday and once on Sunday.
b) Exactly $2n$ members participate in any cycling event.
c) No member may participate in both cycling events of a week.
d) After all cycling events are completed, the number of events where each pair of members meet is the same for all pairs of members.
Prove that after all cycling events are completed, the number of events where each group of three members meet is the same value $t$ for all groups of three members, and that for $n \ge 2$, $t$ is divisible by $n-1$.
1999 CentroAmerican, 1
Suppose that each of the 5 persons knows a piece of information, each piece is different, about a certain event. Each time person $A$ calls person $B$, $A$ gives $B$ all the information that $A$ knows at that moment about the event, while $B$ does not say to $A$ anything that he knew.
(a) What is the minimum number of calls are necessary so that everyone knows about the event?
(b) How many calls are necessary if there were $n$ persons?
1995 Irish Math Olympiad, 4
Consider the following one-person game played on the real line. During the game disks are piled at some of the integer points on the line. To perform a move in the game, the player chooses a point $ j$ at which at least two disks are piled and then takes two disks from the point $ j$ and places one of them at $ j\minus{}1$ and one at $ j\plus{}1$. Initially, $ 2n\plus{}1$ disks are placed at point $ 0$. The player proceeds to perform moves as long as possible. Prove that after $ \frac{1}{6} n(n\plus{}1)(2n\plus{}1)$ moves no further moves will be possible and that at this stage, one disks remains at each of the positions $ \minus{}n,\minus{}n\plus{}1,...,0,...n$.
2006 QEDMO 2nd, 11
On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.
1999 Greece National Olympiad, 4
On a circle are given $n\ge 3$ points. At most, how many parts can the segments with the endpoints at these $n$ points divide the interior of the circle into?
2010 Indonesia TST, 4
Prove that the number $ (\underbrace{9999 \dots 99}_{2005}) ^{2009}$ can be obtained by erasing some digits of $ (\underbrace{9999 \dots 99}_{2008}) ^{2009}$ (both in decimal representation).
[i]Yudi Satria, Jakarta[/i]
2011 Lusophon Mathematical Olympiad, 3
Let $d$ be a positive real number. The scorpion tries to catch the flea on a $10\times 10$ chessboard. The length of the side of each small square of the chessboard is $1$. In this game, the flea and the scorpion move alternately. The flea is always on one of the $121$ vertexes of the chessboard and, in each turn, can jump from the vertex where it is to one of the adjacent vertexes. The scorpion moves on the boundary line of the chessboard, and, in each turn, it can walk along any path of length less than $d$.
At the beginning, the flea is at the center of the chessboard and the scorpion is at a point that he chooses on the boundary line. The flea is the first one to play. The flea is said to [i]escape[/i] if it reaches a point of the boundary line, which the scorpion can't reach in the next turn. Obviously, for big values of $d$, the scorpion has a strategy to prevent the flea's escape. For what values of $d$ can the flea escape? Justify your answer.
1994 Turkey Team Selection Test, 3
All sides and diagonals of a $25$-gon are drawn either red or white. Show that at least $500$ triangles, having all three sides are in same color and having all three vertices from the vertices of the $25$-gon, can be found.
2024 Baltic Way, 9
Let $S$ be a finite set. For a positive integer $n$, we say that a function $f\colon S\to S$ is an [i]$n$-th power[/i] if there exists some function $g\colon S\to S$ such that
\[
f(x) = \underbrace{g(g(\ldots g(x)\ldots))}_{\mbox{\scriptsize $g$ applied $n$ times}}
\]
for each $x\in S$.
Suppose that a function $f\colon S\to S$ is an $n$-th power for each positive integer $n$. Is it necessarily true that $f(f(x)) = f(x)$ for each $x\in S$?
2001 India IMO Training Camp, 3
Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$.
2007 Romania National Olympiad, 4
Given a set $A$ and a function $f: A\rightarrow A$, denote by $f_{1}(A)=f(A)$, $f_{2}(A)=f(f_{1}(A))$, $f_{3}(A)=f(f_{2}(A))$, and so on, ($f_{n}(A)=f(f_{n-1}(A))$, where the notation $f(B)$ means the set $\{ f(x) \ : \ x\in B\}$ of images of points from $B$).
Denote also by $f_{\infty}(A)=f_{1}(A)\cap f_{2}(A)\cap \ldots = \bigcap_{n\geq 1}f_{n}(A)$.
a) Show that if $A$ is finite, then $f(f_{\infty}(A))=f_{\infty}(A)$.
b) Determine if the above is true for $A=\mathbb{N}\times \mathbb{N}$ and the function
\[f\big((m,n)\big)=\begin{cases}(m+1,n) & \mbox{if }n\geq m\geq 1 \\ (0,0) & \mbox{if }m>n \\ (0,n+1) & \mbox{if }n=0. \end{cases}\]