Found problems: 1488
1989 China Team Selection Test, 4
$\forall n \in \mathbb{N}$, $P(n)$ denotes the number of the partition of $n$ as the sum of positive integers (disregarding the order of the parts), e.g. since $4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4$, so $P(4)=5$. "Dispersion" of a partition denotes the number of different parts in that partitation. And denote $q(n)$ is the sum of all the dispersions, e.g. $q(4)=1+2+2+1+1=7$. $n \geq 1$. Prove that
(1) $q(n) = 1 + \sum^{n-1}_{i=1} P(i).$
(2) $1 + \sum^{n-1}_{i=1} P(i) \leq \sqrt{2} \cdot n \cdot P(n)$.
2009 Nordic, 3
The integers $1$, $2$, $3$, $4$, and $5$ are written on a blackboard. It is allowed to wipe out two integers $a$ and $b$ and replace them with $a + b$ and $ab$. Is it possible, by repeating this procedure, to reach a situation where three of the five integers on the blackboard are $2009$?
2012 Canada National Olympiad, 4
A number of robots are placed on the squares of a finite, rectangular grid of squares. A square can hold any number of robots. Every edge of each square of the grid is classified as either passable or impassable. All edges on the boundary of the grid are impassable. You can give any of the commands up, down, left, or right.
All of the robots then simultaneously try to move in the specified direction. If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square. Otherwise, the robot remains on its current square. You can then give another command of up, down, left, or right, then another, for as long as you want. Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square. Prove that you can also give a finite sequence of commands such that all of the robots end up on the same square at the same time.
1995 Hungary-Israel Binational, 4
Consider a convex polyhedron whose faces are triangles. Prove that it is possible to color its edges by either red or blue, in a way that the following property is satisfied: one can travel from any vertex to any other vertex while passing only along red edges, and can also do this while passing only along blue edges.
2009 ITAMO, 1
A flea is initially at the point $(0, 0)$ in the Cartesian plane. Then it makes $n$ jumps. The direction of the jump is taken in a choice of the four cardinal directions. The first step is of length $1$, the second of length $2$, the third of length $4$, and so on. The $n^{th}$-jump is of length $2^{n-1}$. Prove that, if you know the final position flea, then it is possible to uniquely determine its position after each of the $n$ jumps.
2002 Iran MO (2nd round), 2
A rectangle is partitioned into finitely many small rectangles. We call a point a cross point if it belongs to four different small rectangles. We call a segment on the obtained diagram maximal if there is no other segment containing it. Show that the number of maximal segments plus the number of cross points is $3$ more than the number of small rectangles.
1996 IberoAmerican, 3
We have a grid of $k^2-k+1$ rows and $k^2-k+1$ columns, where $k=p+1$ and $p$ is prime. For each prime $p$, give a method to put the numbers 0 and 1, one number for each square in the grid, such that on each row there are exactly $k$ 0's, on each column there are exactly $k$ 0's, and there is no rectangle with sides parallel to the sides of the grid with 0s on each four vertices.
2010 India IMO Training Camp, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
2000 Baltic Way, 7
In a $ 40 \times 50$ array of control buttons, each button has two states: on and off . By touching a button, its state and the states of all buttons in the same row and in the same column are switched. Prove that the array of control buttons may be altered from the all-off state to the all-on state by touching buttons successively, and determine the least number of touches needed to do so.
2007 APMO, 3
Consider $n$ disks $C_{1}; C_{2}; ... ; C_{n}$ in a plane such that for each $1 \leq i < n$, the center of $C_{i}$ is on the circumference of $C_{i+1}$, and the center of $C_{n}$ is on the circumference of $C_{1}$. Define the [i]score[/i] of such an arrangement of $n$ disks to be the number of pairs $(i; j )$ for which $C_{i}$ properly contains $C_{j}$ . Determine the maximum possible score.
2008 Iran MO (2nd Round), 2
We want to choose telephone numbers for a city. The numbers have $10$ digits and $0$ isn’t used in the numbers. Our aim is: We don’t choose some numbers such that every $2$ telephone numbers are different in more than one digit OR every $2$ telephone numbers are different in a digit which is more than $1$. What is the maximum number of telephone numbers which can be chosen? In how many ways, can we choose the numbers in this maximum situation?
1983 IMO Longlists, 37
The points $A_1,A_2, \ldots , A_{1983}$ are set on the circumference of a circle and each is given one of the values $\pm 1$. Show that if the number of points with the value $+1$ is greater than $1789$, then at least $1207$ of the points will have the property that the partial sums that can be formed by taking the numbers from them to any other point, in either direction, are strictly positive.
1994 Hong Kong TST, 2
In a table-tennis tournament of $10$ contestants, any $2$ contestants meet only once.
We say that there is a winning triangle if the following situation occurs: $i$-th contestant defeated the $j$-th contestant, $j$-th contestant defeated the $k$-th contestant, and, $k$-th contestant defeated the $i$-th contestant.
Let, $W_i$ and $L_i $ be respectively the number of games won and lost by the $i$-th contestant.
Suppose, $L_i+W_j\geq 8$ whenever the $j$-th contestant defeats the $i$-th contestant.
Prove that, there are exactly $40$ winning triangles in this tournament.
2005 MOP Homework, 2
Exactly one integer is written in each square of an $n$ by $n$ grid, $n \ge 3$. The sum of all of the numbers in any $2 \times 2$ square is even and the sum of all the numbers in any $3 \times 3$ square is even. Find all $n$ for which the sum of all the numbers in the grid is necessarily even.
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 Finnish National High School Mathematics Competition, 4
There are $110$ points in a unit square. Show that some four of these points reside in a circle whose radius is $1/8.$
2001 Tournament Of Towns, 3
Let $n\ge3$ be an integer. Each row in an $(n-2)\times n$ array consists of the numbers 1,2,...,$n$ in some order, and the numbers in each column are all different. Prove that this array can be expanded into an $n\times n$ array such that each row and each column consists of the numbers 1,2,...,$n$.
1997 South africa National Olympiad, 6
Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)
2006 Estonia Math Open Junior Contests, 7
A solid figure consisting of unit cubes is shown in the picture. Is it possible to exactly fill a cube with these figures if the side length of the cube is
a) 15;
b) 30?
2001 Cono Sur Olympiad, 3
Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Show that one can choose a vertex from each triangle so that the three chosen points determine a triangle each of whose angles is at most $90^\circ$.
2009 Argentina Team Selection Test, 5
There are several contestants at a math olympiad. We say that two contestants $ A$ and $ B$ are [i]indirect friends[/i] if there are contestants $ C_1, C_2, ..., C_n$ such that $ A$ and $ C_1$ are friends, $ C_1$ and $ C_2$ are friends, $ C_2$ and $ C_3$ are friends, ..., $ C_n$ and $ B$ are friends. In particular, if $ A$ and $ B$ are friends themselves, they are [i]indirect friends[/i] as well.
Some of the contestants were friends before the olympiad. During the olympiad, some contestants make new friends, so that after the olympiad every contestant has at least one friend among the other contestants. We say that a contestant is [i]special[/i] if, after the olympiad, he has exactly twice as indirect friends as he had before the olympiad.
Prove that the number of special contestants is less or equal than $ \frac{2}{3}$ of the total number of contestants.
2002 China Girls Math Olympiad, 2
There are $ 3n, n \in \mathbb{Z}^\plus{}$ girl students who took part in a summer camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the $ 3n$ students had just one time to be on duty on the same day.
(1) When $ n\equal{}3,$ is there any arrangement satisfying the requirement above. Prove yor conclusion.
(2) Prove that $ n$ is an odd number.
2001 All-Russian Olympiad, 4
Some towns in a country are connected by two–way roads, so that for any two towns there is a unique path along the roads connecting them. It is known that there is exactly 100 towns which are directly connected to only one town. Prove that we can construct 50 new roads in order to obtain a net in which every two towns will be connected even if one road gets closed.
2013 Rioplatense Mathematical Olympiad, Level 3, 3
A division of a group of people into various groups is called $k$-regular if the number of groups is less or equal to $k$ and two people that know each other are in different groups.
Let $A$, $B$, and $C$ groups of people such that there are is no person in $A$ and no person in $B$ that know each other. Suppose that the group $A \cup C$ has an $a$-regular division and the group $B \cup C$ has a $b$-regular division.
For each $a$ and $b$, determine the least possible value of $k$ for which it is guaranteed that the group $A \cup B \cup C$ has a $k$-regular division.
1998 China Team Selection Test, 2
$n \geq 5$ football teams participate in a round-robin tournament. For every game played, the winner receives 3 points, the loser receives 0 points, and in the event of a draw, both teams receive 1 point. The third-from-bottom team has fewer points than all the teams ranked before it, and more points than the last 2 teams; it won more games than all the teams before it, but fewer games than the 2 teams behind it. Find the smallest possible $n$.