Found problems: 60
2004 Romania Team Selection Test, 9
Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements.
Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.
1999 Brazil National Olympiad, 5
There are $n$ football teams in [i]Tumbolia[/i]. A championship is to be organised in which each team plays against every other team exactly once. Ever match takes place on a sunday and each team plays at most one match each sunday. Find the least possible positive integer $m_n$ for which it is possible to set up a championship lasting $m_n$ sundays.
2019 Hong Kong TST, 5
Is it is possible to choose 24 distinct points in the space such that no three of them lie on the same line and choose 2019 distinct planes in a way that each plane passes through at least 3 of the chosen points and each triple belongs to one of the chosen planes?
2017 USA TSTST, 2
Ana and Banana are playing a game. First Ana picks a word, which is defined to be a nonempty sequence of capital English letters. (The word does not need to be a valid English word.) Then Banana picks a nonnegative integer $k$ and challenges Ana to supply a word with exactly $k$ subsequences which are equal to Ana's word. Ana wins if she is able to supply such a word, otherwise she loses.
For example, if Ana picks the word "TST", and Banana chooses $k=4$, then Ana can supply the word "TSTST" which has 4 subsequences which are equal to Ana's word.
Which words can Ana pick so that she wins no matter what value of $k$ Banana chooses?
(The subsequences of a string of length $n$ are the $2^n$ strings which are formed by deleting some of its characters, possibly all or none, while preserving the order of the remaining characters.)
[i]Proposed by Kevin Sun
2012 Tuymaada Olympiad, 1
Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy?
[i]Proposed by A. Golovanov[/i]
2016 Israel Team Selection Test, 4
A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?
1990 IMO Longlists, 31
Let $S = \{1, 2, \ldots, 1990\}$. A $31$-element subset of $S$ is called "good" if the sum of its elements is divisible by $5$. Find the number of good subsets of $S.$
2006 USAMO, 5
A mathematical frog jumps along the number line. The frog starts at $1$, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n + 2^{m_n+1}$ where $2^{m_n}$ is the largest power of $2$ that is a factor of $n.$ Show that if $k \geq 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$
2004 Germany Team Selection Test, 3
We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black.
Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black?
[It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]
1998 Finnish National High School Mathematics Competition, 2
There are $11$ members in the competetion committee. The problem set is kept in a safe having several locks.
The committee members have been provided with keys in such a way that every six members can open the safe, but no fi ve members can do that.
What is the smallest possible number of locks, and how many keys are needed in that case?
2018 Tuymaada Olympiad, 6
The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values.
[i]Proposed by A. Golovanov[/i]
2018 Slovenia Team Selection Test, 1
Let $n$ be a positive integer. On the table, we have $n^2$ ornaments in $n$ different colours, not necessarily $n$ of each colour. Prove that we can hang the ornaments on $n$ Christmas trees in such a way that there are exactly $n$ ornaments on each tree and the ornaments on every tree are of at most $2$ different colours.
1991 Bundeswettbewerb Mathematik, 2
In the space there are 8 points that no four of them are in the plane. 17 of the connecting segments are coloured blue and the other segments are to be coloured red. Prove that this colouring will create at least four triangles. Prove also that four cannot be subsituted by five.
Remark: Blue triangles are those triangles whose three edges are coloured blue.
1991 IMTS, 5
Two people, $A$ and $B$, play the following game with a deck of 32 cards. With $A$ starting, and thereafter the players alternating, each player takes either 1 card or a prime number of cards. Eventually all of the cards are chosen, and the person who has none to pick up is the loser. Who will win the game if they both follow optimal strategy?
1990 IMO Longlists, 40
Given three letters $X, Y, Z$, we can construct letter sequences arbitrarily, such as $XZ, ZZYXYY, XXYZX$, etc. For any given sequence, we can perform following operations:
$T_1$: If the right-most letter is $Y$, then we can add $YZ$ after it, for example, $T_1(XYZXXY) =
(XYZXXYYZ).$
$T_2$: If The sequence contains $YYY$, we can replace them by $Z$, for example, $T_2(XXYYZYYYX) =
(XXYYZZX).$
$T_3$: We can replace $Xp$ ($p$ is any sub-sequence) by $XpX$, for example, $T_3(XXYZ) = (XXYZX).$
$T_4$: In a sequence containing one or more $Z$, we can replace the first $Z$ by $XY$, for example,
$T_4(XXYYZZX) = (XXYYXYZX).$
$T_5$: We can replace any of $XX, YY, ZZ$ by $X$, for example, $T_5(ZZYXYY) = (XYXX)$ or $(XYXYY)$ or $(ZZYXX).$
Using above operations, can we get $XYZZ$ from $XYZ \ ?$
2001 All-Russian Olympiad, 2
In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and columns are equal.)
2016 JBMO TST - Turkey, 2
A and B plays a game on a pyramid whose base is a $2016$-gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the $k$ colors such that none of the sides with a common vertex have the same color. If A starts the game, find the minimal value of $k$ for which $B$ can guarantee that all sides are colored.
2014 JBMO Shortlist, 3
For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?
2004 Germany Team Selection Test, 3
We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules:
(a) We can add an arbitrary integer to the numbers at two opposite vertices.
(b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle.
(c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers.
Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)
2012 Tuymaada Olympiad, 1
Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy?
[i]Proposed by A. Golovanov[/i]
2004 Brazil National Olympiad, 2
Determine all values of $n$ such that it is possible to divide a triangle in $n$ smaller triangles such that there are not three collinear vertices and such that each vertex belongs to the same number of segments.
2016 Israel Team Selection Test, 3
On each square of an $n$x$n$ board sleeps a dragon. Two dragons are called neighbors if their squares have a side in common. Each turn, Minnie wakes up a dragon which has a living neighbor and Max directs it towards one of its living neighbors. The dragon than breathes fire on that neighbor and destroys it, and then goes back to sleep.
Minnie's goal is to minimize the snoring of the dragons and leave as few living dragons as possible. Max is a member of PETD (People for the Ethical Treatment of Dragons), and he wants to save as many dragons as he can.
How many dragons will stay alive at the end if
1. $n=4$?
2. $n=5$?
2004 Canada National Olympiad, 2
How many ways can $ 8$ mutually non-attacking rooks be placed on the $ 9\times9$ chessboard (shown here) so that all $ 8$ rooks are on squares of the same color?
(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.)
[asy]unitsize(3mm);
defaultpen(white);
fill(scale(9)*unitsquare,black);
fill(shift(1,0)*unitsquare);
fill(shift(3,0)*unitsquare);
fill(shift(5,0)*unitsquare);
fill(shift(7,0)*unitsquare);
fill(shift(0,1)*unitsquare);
fill(shift(2,1)*unitsquare);
fill(shift(4,1)*unitsquare);
fill(shift(6,1)*unitsquare);
fill(shift(8,1)*unitsquare);
fill(shift(1,2)*unitsquare);
fill(shift(3,2)*unitsquare);
fill(shift(5,2)*unitsquare);
fill(shift(7,2)*unitsquare);
fill(shift(0,3)*unitsquare);
fill(shift(2,3)*unitsquare);
fill(shift(4,3)*unitsquare);
fill(shift(6,3)*unitsquare);
fill(shift(8,3)*unitsquare);
fill(shift(1,4)*unitsquare);
fill(shift(3,4)*unitsquare);
fill(shift(5,4)*unitsquare);
fill(shift(7,4)*unitsquare);
fill(shift(0,5)*unitsquare);
fill(shift(2,5)*unitsquare);
fill(shift(4,5)*unitsquare);
fill(shift(6,5)*unitsquare);
fill(shift(8,5)*unitsquare);
fill(shift(1,6)*unitsquare);
fill(shift(3,6)*unitsquare);
fill(shift(5,6)*unitsquare);
fill(shift(7,6)*unitsquare);
fill(shift(0,7)*unitsquare);
fill(shift(2,7)*unitsquare);
fill(shift(4,7)*unitsquare);
fill(shift(6,7)*unitsquare);
fill(shift(8,7)*unitsquare);
fill(shift(1,8)*unitsquare);
fill(shift(3,8)*unitsquare);
fill(shift(5,8)*unitsquare);
fill(shift(7,8)*unitsquare);
draw(scale(9)*unitsquare,black);[/asy]
2001 Bundeswettbewerb Mathematik, 1
10 vertices of a regular 100-gon are coloured red and ten other (distinct) vertices are coloured blue. Prove that there is at least one connection edge (segment) of two red which is as long as the connection edge of two blue points.
[hide="Hint"]Possible approaches are pigeon hole principle, proof by contradiction, consider turns (bijective congruent mappings) which maps red in blue points.
[/hide]
1992 China Team Selection Test, 1
16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.