Found problems: 1488
1990 USAMO, 3
Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has 19. Prove that for any odd integer $n \geq 1$, there is a way to number each of the 33 beads with an integer from the sequence \[ \{ n, n+1, n+2, \dots, n+32 \} \] so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace'' is viewed as a circle in which each bead is adjacent to two other beads.)
2004 Greece National Olympiad, 4
Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$
If no element of $M$ is equal to the sum of any two elements of $M,$
find the least value that the greatest element of $M$ can take.
1984 IMO Longlists, 21
$(1)$ Start with $a$ white balls and $b$ black balls.
$(2)$ Draw one ball at random.
$(3)$ If the ball is white, then stop. Otherwise, add two black balls and go to step $2$.
Let $S$ be the number of draws before the process terminates. For the cases $a = b = 1$ and $a = b = 2$ only, find $a_n = P(S = n), b_n = P(S \le n), \lim_{n\to\infty} b_n$, and the expectation value of the number of balls drawn: $E(S) =\displaystyle\sum_{n\ge1} na_n.$
2007 Brazil National Olympiad, 6
Given real numbers $ x_1 < x_2 < \ldots < x_n$ such that every real number occurs at most two times among the differences $ x_j \minus{} x_i$, $ 1\leq i < j \leq n$, prove that there exists at least $ \lfloor n/2\rfloor$ real numbers that occurs exactly one time among such differences.
2001 USA Team Selection Test, 4
There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does [i]not[/i] necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.
2010 Tuymaada Olympiad, 4
(I'll skip over the whole "dressing" of the graph in cities and flights [color=#FF0000][Mod edit: Shu has posted the "dressed-up" version below][/color])
For an ordinary directed graph, show that there is a subset A of vertices such that:
$1.$ There are no edges between the vertices of A.
$2.$ For any vertex $v$, there is either a direct way from $v$ to a vertex in A, or a way passing through only one vertex and ending in A (like $v$ ->$v'$-> $a$, where $a$ is a vertex in A)
2000 Italy TST, 4
On a mathematical competition $ n$ problems were given. The final results showed that:
(i) on each problem, exactly three contestants scored $ 7$ points;
(ii) for each pair of problems, exactly one contestant scored $ 7$ points on both problems.
Prove that if $ n \geq 8$, then there is a contestant who got $ 7$ points on each problem. Is this statement necessarily true if $ n \equal{} 7$?
2003 Baltic Way, 8
There are $2003$ pieces of candy on a table. Two players alternately make moves. A move consists of eating one candy or half of the candies on the table (the “lesser half” if there are an odd number of candies). At least one candy must be eaten at each move. The loser is the one who eats the last candy. Which player has a winning strategy?
2010 Argentina Team Selection Test, 4
Two players, $A$ and $B$, play a game on a board which is a rhombus of side $n$ and angles of $60^{\circ}$ and $120^{\circ}$, divided into $2n^2$ equilateral triangles, as shown in the diagram for $n=4$.
$A$ uses a red token and $B$ uses a blue token, which are initially placed in cells containing opposite corners of the board (the $60^{\circ}$ ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started.
If $A$ starts the game, determine which player has a winning strategy.
2007 Estonia Math Open Junior Contests, 9
In an exam with k questions, n students are taking part. A student fails the exam
if he answers correctly less than half of all questions. Call a question easy if more than half of all students answer it correctly. For which pairs (k, n) of positive integers is it possible that
(a) all students fail the exam although all questions are easy;
(b) no student fails the exam although no question is easy?
2012 Philippine MO, 5
There are exactly $120$ Twitter subscribers from National Science High School. Statistics show that each of $10$ given celebrities has at least $85$ followers from National Science High School. Prove that there must be two students such that each of the $10$ celebrities is being followed in Twitter by at least one of these students.
2006 CHKMO, 1
On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.
2010 China Team Selection Test, 3
Let $A$ be a finite set, and $A_1,A_2,\cdots, A_n$ are subsets of $A$ with the following conditions:
(1) $|A_1|=|A_2|=\cdots=|A_n|=k$, and $k>\frac{|A|}{2}$;
(2) for any $a,b\in A$, there exist $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$ such that
$a,b\in A_r\cap A_s\cap A_t$;
(3) for any integer $i,j\, (1\leq i<j\leq n)$, $|A_i\cap A_j|\leq 3$.
Find all possible value(s) of $n$ when $k$ attains maximum among all possible systems $(A_1,A_2,\cdots, A_n,A)$.
2009 Tournament Of Towns, 1
Each of $10$ identical jars contains some milk, up to $10$ percent of its capacity. At any time, we can tell the precise amount of milk in each jar. In a move, we may pour out an exact amount of milk from one jar into each of the other $9$ jars, the same amount in each case. Prove that we can have the same amount of milk in each jar after at most $10$ moves.
[i](4 points)[/i]
1998 Bulgaria National Olympiad, 1
Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of $0$ and $1$ of length $2n+2$ with the following property: any sequence of $0$ and $1$ of length $2n+2$ coincides with some of these $k$ sequences in at least $n+2$ positions.
2009 Argentina Team Selection Test, 1
On a $ 50 \times 50$ board, the centers of several unit squares are colored black. Find the maximum number of centers that can be colored black in such a way that no three black points form a right-angled triangle.
2006 Canada National Olympiad, 4
Consider a round-robin tournament with $2n+1$ teams, where each team plays each other team exactly one. We say that three teams $X,Y$ and $Z$, form a [i]cycle triplet [/i] if $X$ beats $Y$, $Y$ beats $Z$ and $Z$ beats $X$. There are no ties.
a)Determine the minimum number of cycle triplets possible.
b)Determine the maximum number of cycle triplets possible.
1996 India National Olympiad, 6
There is a $2n \times 2n$ array (matrix) consisting of $0's$ and $1's$ and there are exactly $3n$ zeroes. Show that it is possible to remove all the zeroes by deleting some $n$ rows and some $n$ columns.
2000 Moldova Team Selection Test, 7
Suppose that $ p_1,p_2,p_3,q_1,q_2,q_3$ are six points in the plane and that the distance between $ p_i$ and $ q_j$ ($ i,j \equal{} 1,2,3$) is $ i \plus{} j$. Show that the six points are collinear.
2002 China Team Selection Test, 1
$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$.
(1) Prove that we can divide all the rational points into $ 100$ pairwisely non-intersecting point sets with infinity elements. If for any line on the plane, there are two rational points on it, then it passes through all the $ 100$ sets.
(2) Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.
2010 India IMO Training Camp, 6
Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to
\[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]
2007 Estonia National Olympiad, 2
A 3-dimensional chess board consists of $ 4 \times 4 \times 4$ unit cubes. A rook can step from any unit cube K to any other unit cube that has a common face with K. A bishop can step from any unit cube K to any other unit cube that has a common edge with K, but does not have a common face. One move of both a rook and a bishop consists of an arbitrary positive number of consecutive steps in the same direction. Find the average number of possible moves for either piece, where the average is taken over all possible starting cubes K.
2011 China Team Selection Test, 3
Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$.
Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.
2007 CHKMO, 1
Let M be a subst of {1,2,...,2006} with the following property: For any three elements x,y and z (x<y<z) of M, x+y does not divide z. Determine the largest possible size of M. Justify your claim.
2006 MOP Homework, 1
Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can not use a needle to peer through the tiling? Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can use a needle to through the tiling? What if it is a $6 \times 6$ board?