Found problems: 1488
1992 China National Olympiad, 2
Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)
2011 Korea - Final Round, 3
There is a chessboard with $m$ columns and $n$ rows. In each blanks, an integer is given. If a rectangle $R$ (in this chessboard) has an integer $h$ satisfying the following two conditions, we call $R$ as a 'shelf'.
(i) All integers contained in $R$ are bigger than $h$.
(ii) All integers in blanks, which are not contained in $R$ but meet with $R$ at a vertex or a side, are not bigger than $h$.
Assume that all integers are given to make shelves as much as possible. Find the number of shelves.
1970 IMO Longlists, 55
A turtle runs away from an UFO with a speed of $0.2 \ m/s$. The UFO flies $5$ meters above the ground, with a speed of $20 \ m/s$. The UFO's path is a broken line, where after flying in a straight path of length $\ell$ (in meters) it may turn through for any acute angle $\alpha$ such that $\tan \alpha < \frac{\ell}{1000}$. When the UFO's center approaches within $13$ meters of the turtle, it catches the turtle. Prove that for any initial position the UFO can catch the turtle.
2014 IberoAmerican, 3
$2014$ points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of the $2014$ points, the sum of the numbers written on their sides is less or equal than $1$. Find the maximum possible value for the sum of all the written numbers.
1997 Canada National Olympiad, 2
The closed interval $A = [0, 50]$ is the union of a finite number of closed intervals, each of length $1$. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length greater than $25$.
Note: For reals $a\le b$, the closed interval $[a, b] := \{x\in \mathbb{R}:a\le x\le b\}$ has length $b-a$; disjoint intervals have [i]empty [/i]intersection.
2004 Postal Coaching, 4
In how many ways can a $2\times n$ grid be covered by
(a) 2 monominoes and $n-1$ dominoes
(b) 4 monominoes and $n-2$ dominoes.
1991 Romania Team Selection Test, 3
Prove the following identity for every $ n\in N$:
$ \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}$
1998 Finnish National High School Mathematics Competition, 5
$15\times 36$-checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$
Tiles are supposed to cover whole squares of the board and be non-overlapping.
What is the maximum number of squares to be covered?
1991 IMTS, 3
On a 8 x 8 board we place $n$ dominoes, each covering two adjacent squares, so that no more dominoes can be placed on the remaining squares. What is the smallest value of $n$ for which the above statement is true?
2011 Lusophon Mathematical Olympiad, 3
Consider a sequence of equilateral triangles $T_{n}$ as represented below:
[asy]
defaultpen(linewidth(0.8));size(350);
real r=sqrt(3);
path p=origin--(2,0)--(1,sqrt(3))--cycle;
int i,j,k;
for(i=1; i<5; i=i+1) {
for(j=0; j<i; j=j+1) {
for(k=0; k<j; k=k+1) {
draw(shift(5*i-5+(i-2)*(i-1)*1,0)*shift(2(j-k)+k, k*r)*p);
}}}[/asy]
The length of the side of the smallest triangles is $1$. A triangle is called a delta if its vertex is at the top; for example, there are $10$ deltas in $T_{3}$. A delta is said to be perfect if the length of its side is even. How many perfect deltas are there in $T_{20}$?
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 $C_{n+2}-2C_{n+1}+C_n$
1989 IMO Longlists, 76
Poldavia is a strange kingdom. Its currency unit is the bourbaki and there exist only two types of coins: gold ones and silver ones. Each gold coin is worth $ n$ bourbakis and each silver coin is worth $ m$ bourbakis ($ n$ and $ m$ are positive integers). Using gold and silver coins, it is possible to obtain sums such as 10000 bourbakis, 1875 bourbakis, 3072 bourbakis, and so on. But Poldavia’s monetary system is not as strange as it seems:
[b](a)[/b] Prove that it is possible to buy anything that costs an integral number of bourbakis, as long as one can receive change.
[b](b)[/b] Prove that any payment above $ mn\minus{}2$ bourbakis can be made without the need to receive change.
1996 Greece National Olympiad, 4
Find the number of functions $f : \{1, 2, . . . , n\} \to \{1995, 1996\}$ such that $f(1) + f(2) + ... + f(1996)$ is odd.
2001 Canada National Olympiad, 2
There is a board numbered $-10$ to $10$. Each square is coloured either red or white, and the sum of the numbers on the red squares is $n$. Maureen starts with a token on the square labeled $0$. She then tosses a fair coin ten times. Every time she flips heads, she moves the token one square to the right. Every time she flips tails, she moves the token one square to the left. At the end of the ten flips, the probability that the token finishes on a red square is a rational number of the form $\frac a b$. Given that $a + b = 2001$, determine the largest possible value for $n$.
2008 Tournament Of Towns, 3
There are $N$ piles each consisting of a single nut. Two players in turns play the following game. At each move, a player combines two piles that contain coprime numbers of nuts into a new pile. A player who can not make a move, loses. For every $N > 2$ determine which of the players, the first or the second, has a winning strategy.
2001 Tournament Of Towns, 3
On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships have returned to port, how many meetings of two ships have taken place?
1994 Canada National Olympiad, 3
$25$ men sit around a circular table. Every hour there is a vote, and each must respond [i]yes [/i]or [i]no[/i]. Each man behaves as follows: on the $n^{\text{th}}$, vote if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)^{\text{th}}$ vote as on the $n^{\text{th}}$ vote; but if his response is different from that of both his neighbours on the $n^{\text{th}}$ vote, then his response on the $(n+1)^{\text{th}}$ vote will be different from his response on the $n^{\text{th}}$ vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.
1999 Turkey MO (2nd round), 6
We wish to find the sum of $40$ given numbers utilizing $40$ processors. Initially, we have the number $0$ on the screen of each processor. Each processor adds the number on its screen with a number entered directly (only the given numbers could be entered directly to the processors) or transferred from another processor in a unit time. Whenever a number is transferred from a processor to another, the former processor resets. Find the least time needed to find the desired sum.
2010 Albania Team Selection Test, 5
[b]a)[/b] Let's consider a finite number of big circles of a sphere that do not pass all from a point. Show that there exists such a point that is found only in two of the circles. (With big circle we understand the circles with radius equal to the radius of the sphere.)
[b]b)[/b] Using the result of part $a)$ show that, for a set of $n$ points in a plane, that are not all in a line, there exists a line that passes through only two points of the given set.
2017 Bundeswettbewerb Mathematik, 1
The numbers $1,2,3,\dots,2017$ are on the blackboard. Amelie and Boris take turns removing one of those until only two numbers remain on the board. Amelie starts. If the sum of the last two numbers is divisible by $8$, then Amelie wins. Else Boris wins. Who can force a victory?
2002 India IMO Training Camp, 2
Show that there is a set of $2002$ consecutive positive integers containing exactly $150$ primes. (You may use the fact that there are $168$ primes less than $1000$)
2008 Bundeswettbewerb Mathematik, 4
On a bookcase there are $ n \geq 3$ books side by side by different authors. A librarian considers the first and second book from left and exchanges them iff they are not alphabetically sorted. Then he is doing the same operation with the second and third book from left etc. Using this procedure he iterates through the bookcase three times from left to right. Considering all possible initial book configurations how many of them will then be alphabetically sorted?
2010 Contests, 3
Let $a_0, a_1, \ldots, a_9$ and $b_1 , b_2, \ldots,b_9$ be positive integers such that $a_9<b_9$ and $a_k \neq b_k, 1 \leq k \leq 8.$ In a cash dispenser/automated teller machine/ATM there are $n\geq a_9$ levs (Bulgarian national currency) and for each $1 \leq i \leq 9$ we can take $a_i$ levs from the ATM (if in the bank there are at least $a_i$ levs). Immediately after that action the bank puts $b_i$ levs in the ATM or we take $a_0$ levs. If we take $a_0$ levs from the ATM the bank doesn’t put any money in the ATM. Find all possible positive integer values of $n$ such that after finite number of takings money from the ATM there will be no money in it.
2014 Contests, 1
In a plane, 2014 lines are distributed in 3 groups. in every group all the lines are parallel between themselves. What is the maximum number of triangles that can be formed, such that every side of such triangle lie on one of the lines?
2010 Argentina Team Selection Test, 6
Suppose $a_1, a_2, ..., a_r$ are integers with $a_i \geq 2$ for all $i$ such that $a_1 + a_2 + ... + a_r = 2010$.
Prove that the set $\{1,2,3,...,2010\}$ can be partitioned in $r$ subsets $A_1, A_2, ..., A_r$ each with $a_1, a_2, ..., a_r$ elements respectively, such that the sum of the numbers on each subset is divisible by $2011$.
Decide whether this property still holds if we replace $2010$ by $2011$ and $2011$ by $2012$ (that is, if the set to be partitioned is $\{1,2,3,...,2011\}$).