Found problems: 1488
2002 China Team Selection Test, 3
Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that
- One person should book at most one admission ticket for one match;
- At most one match was same in the tickets booked by every two persons;
- There was one person who booked six tickets.
How many tickets did those football fans book at most?
2013 ELMO Shortlist, 6
A $4\times4$ grid has its 16 cells colored arbitrarily in three colors. A [i]swap[/i] is an exchange between the colors of two cells. Prove or disprove that it always takes at most three swaps to produce a line of symmetry, regardless of the grid's initial coloring.
[i]Proposed by Matthew Babbitt[/i]
1970 IMO Longlists, 53
A square $ABCD$ is divided into $(n - 1)^2$ congruent squares, with sides parallel to the sides of the given square. Consider the grid of all $n^2$ corners obtained in this manner. Determine all integers $n$ for which it is possible to construct a non-degenerate parabola with its axis parallel to one side of the square and that passes through exactly $n$ points of the grid.
2013 China Western Mathematical Olympiad, 4
There are $n$ coins in a row, $n\geq 2$. If one of the coins is head, select an odd number of consecutive coins (or even 1 coin) with the one in head on the leftmost, and then flip all the selected coins upside down simultaneously. This is a $move$. No move is allowed if all $n$ coins are tails.
Suppose $m-1$ coins are heads at the initial stage, determine if there is a way to carry out $ \lfloor\frac {2^m}{3}\rfloor $ moves
1988 IMO Longlists, 50
Prove that the numbers $A,B$ and $C$ are equal, where:
- $A=$ number of ways that we can cover a $2 \times n$ rectangle with $2 \times 1$ retangles.
- $B=$ number of sequences of ones and twos that add up to $n$
- $C= \sum^m_{k=0} \binom{m + k}{2 \cdot k}$ if $n = 2 \cdot m,$ and
- $C= \sum^m_{k=0} \binom{m + k + 1}{2 \cdot k + 1}$ if $n = 2 \cdot m + 1.$
1995 Turkey Team Selection Test, 2
Let $n$ be a positive integer. Find the number of permutations $\sigma$ of the set $\{1, 2, ..., n\}$ such that $\sigma(j) \geq j$ holds for exactly two values of $j$.
1986 China National Olympiad, 5
Given a sequence $1,1,2,2,3,3,\ldots,1986,1986$, determine, with proof, if we can rearrange the sequence
so that for any integer $1\le k \le 1986$ there are exactly $k$ numbers between the two “$k$”s.
2014 Greece Team Selection Test, 4
Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.
1997 Pre-Preparation Course Examination, 6
A building has some rooms and there is one or more than one doors between the rooms. We know that we can go from each room to another one. Two rooms $E,S$ has been labeled, and the room $S$ has exactly one door. Someone is in the room $S$ and wants to move to the room $E$.
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A "[i]way[/i]" $P$ for moving between the rooms is an infinite sequence of $L$ and $R$. We say that someone moves according to the "[i]way[/i]" $P$, if he start moving from the room $S$, and after passing the $n$'th door, if $P_n$ is $R$, then he goes to the first door which is in the right side, and if $P_n$ is $L$, then he goes to the first door which is in the left side (obviously, if some room has exactly one door, then there is no difference between $L$ and $R$), and when he arrives to the room $E$, he stops moving. Prove that there exists a "[i]way[/i]" such that if the person move according to it, then he can arrive to the room $E$ of any building.
2003 IberoAmerican, 1
Let $M=\{1,2,\dots,49\}$ be the set of the first $49$ positive integers. Determine the maximum integer $k$ such that the set $M$ has a subset of $k$ elements such that there is no $6$ consecutive integers in such subset. For this value of $k$, find the number of subsets of $M$ with $k$ elements with the given property.
2012 China Girls Math Olympiad, 4
There is a stone at each vertex of a given regular $13$-gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the $13$-gon.
1986 Canada National Olympiad, 2
A Mathlon is a competition in which there are $M$ athletic events. Such a competition was held in which only $A$, $B$, and $C$ participated. In each event $p_1$ points were awarded for first place, $p_2$ for second and $p_3$ for third, where $p_1 > p_2 > p_3 > 0$ and $p_1$, $p_2$, $p_3$ are integers. The final scores for $A$ was 22, for $B$ was 9 and for $C$ was also 9. $B$ won the 100 metres. What is the value of $M$ and who was second in the high jump?
2012 Vietnam National Olympiad, 4
Let $n$ be a natural number. There are $n$ boys and $n$ girls standing in a line, in any arbitrary order. A student $X$ will be eligible for receiving $m$ candies, if we can choose two students of opposite sex with $X$ standing on either side of $X$ in $m$ ways. Show that the total number of candies does not exceed $\frac 13n(n^2-1).$
1984 USAMO, 4
A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.
1996 Kurschak Competition, 3
Let $n$ and $k$ be arbitrary non-negative integers. Suppose we have drawn $2kn+1$ (different) diagonals of a convex $n$-gon. Show that there exists a broken line formed by $2k+1$ of these diagonals that passes through no point more than once. Prove also that this is not necessarily true when we draw only $kn$ diagonals.
2006 Bulgaria Team Selection Test, 3
[b] Problem 6.[/b] Let $m\geq 5$ and $n$ are given natural numbers, and $M$ is regular $2n+1$-gon. Find the number of the convex $m$-gons with vertices among the vertices of $M$, who have at least one acute angle.
[i]Alexandar Ivanov[/i]
2012 Olympic Revenge, 3
Let $G$ be a finite graph. Prove that one can partition $G$ into two graphs $A \cup B=G$ such that if we erase all edges conecting a vertex from $A$ to a vertex from $B$, each vertex of the new graph has even degree.
1993 China Team Selection Test, 3
A graph $G=(V,E)$ is given. If at least $n$ colors are required to paints its vertices so that between any two same colored vertices no edge is connected, then call this graph ''$n-$colored''. Prove that for any $n \in \mathbb{N}$, there is a $n-$colored graph without triangles.
1990 Polish MO Finals, 3
In a tournament, every two of the $n$ players played exactly one match with each other (no
draws). Prove that it is possible either
(i) to partition the league in two groups $A$ and $B$ such that everybody in $A$ defeated everybody in $B$; or
(ii) to arrange all the players in a chain $x_1, x_2, . . . , x_n, x_1$ in such a way that each player defeated his successor.
2010 Moldova Team Selection Test, 4
In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
2008 Finnish National High School Mathematics Competition, 4
Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches.
What is the largest possible number of matches?
2014 Romania Team Selection Test, 2
Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times. Let $b_n$ be the number of words of length $n$, formed with letters from $B$, in which appear all the letters from $B$, each an odd number of times. Compute $\frac{b_n}{a_n}$.
1999 Mediterranean Mathematics Olympiad, 2
A plane figure of area $A > n$ is given, where $n$ is a positive integer. Prove that
this figure can be placed onto a Cartesian plane so that it covers at least $n+1$
points with integer coordinates.
2014 India IMO Training Camp, 3
In how many ways rooks can be placed on a $8$ by $8$ chess board such that every row and every column has at least one rook?
(Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them)
Kvant 2019, M2587
In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?