Found problems: 1800
2014 Taiwan TST Round 1, 2
Determine whether there exist ten sets $A_1$, $A_2$, $\dots$, $A_{10}$ such that
(i) each set is of the form $\{a,b,c\}$, where $a \in \{1,2,3\}$, $b \in \{4,5,6\}$, $c \in \{7,8,9\}$,
(ii) no two sets are the same,
(iii) if the ten sets are arranged in a circle $(A_1, A_2, \dots, A_{10})$, then any two adjacent sets have no common element, but any two non-adjacent sets intersect. (Note: $A_{10}$ is adjacent to $A_1$.)
2006 Moldova National Olympiad, 11.8
Given an alfabet of $n$ letters. A sequence of letters such that between any 2 identical letters there are no 2 identical letters is called a [i]word[/i].
a) Find the maximal possible length of a [i]word[/i].
b) Find the number of the [i]words[/i] of maximal length.
2009 China Team Selection Test, 1
Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$
2008 German National Olympiad, 5
Inside a square of sidelength $ 1$ there are finitely many disks that are allowed to overlap. The sum of all circumferences is $ 10$. Show that there is a line intersecting or touching at least $ 4$ disks.
2004 Iran MO (3rd Round), 2
$A$ is a compact convex set in plane. Prove that there exists a point $O \in A$, such that for every line $XX'$ passing through $O$, where $X$ and $X'$ are boundary points of $A$, then
\[ \frac12 \leq \frac {OX}{OX'} \leq 2.\]
2007 China Team Selection Test, 2
Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segments $ P_{i}P_{j}$ and $ P_{u}P_{v},$ the number of triangles having $ P_{i}P_{j}$ as side and the number of triangles having $ P_{u}P_{v}$ as side are the same in $ S.$ Find the least $ n$ such that in $ S$ there exist two triangles, the vertices of each triangle having the same color.
2010 Indonesia TST, 4
For each positive integer $ n$, define $ f(n)$ as the number of digits $ 0$ in its decimal representation. For example, $ f(2)\equal{}0$, $ f(2009)\equal{}2$, etc. Please, calculate \[ S\equal{}\sum_{k\equal{}1}^{n}2^{f(k)},\] for $ n\equal{}9,999,999,999$.
[i]Yudi Satria, Jakarta[/i]
2011 Cono Sur Olympiad, 2
The numbers $1$ through $4^{n}$ are written on a board. In each step, Pedro erases two numbers $a$ and $b$ from the board, and writes instead the number $\frac{ab}{\sqrt{2a^2+2b^2}}$. Pedro repeats this procedure until only one number remains. Prove that this number is less than $\frac{1}{n}$, no matter what numbers Pedro chose in each step.
2011 All-Russian Olympiad, 3
There are 999 scientists. Every 2 scientists are both interested in exactly 1 topic and for each topic there are exactly 3 scientists that are interested in that topic. Prove that it is possible to choose 250 topics such that every scientist is interested in at most 1 theme.
[i]A. Magazinov[/i]
2010 Iran MO (3rd Round), 4
suppose that $\mathcal F\subseteq X^{(K)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B$ and $C$ we have $A\cap B \not\subset C$.
a)(10 points) Prove that :
\[|\mathcal F|\le \dbinom{k}{\lfloor\frac{k}{2}\rfloor}+1\]
b)(15 points) if elements of $\mathcal F$ do not necessarily have $k$ elements, with the above conditions show that:
\[|\mathcal F|\le \dbinom{n}{\lceil\frac{n-2}{3}\rceil}+2\]
2006 Baltic Way, 7
A photographer took some pictures at a party with $10$ people. Each of the $45$ possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken?
2010 IberoAmerican, 1
There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?
1998 Iran MO (2nd round), 3
If $A=(a_1,\cdots,a_n)$ , $B=(b_1,\cdots,b_n)$ be $2$ $n-$tuple that $a_i,b_i=0 \ or \ 1$ for $i=1,2,\cdots,n$, we define $f(A,B)$ the number of $1\leq i\leq n$ that $a_i\ne b_i$.
For instance, if $A=(0,1,1)$ , $B=(1,1,0)$, then $f(A,B)=2$.
Now, let $A=(a_1,\cdots,a_n)$ , $B=(b_1,\cdots,b_n)$ , $C=(c_1,\cdots,c_n)$ be 3 $n-$tuple, such that for $i=1,2,\cdots,n$, $a_i,b_i,c_i=0 \ or \ 1$ and $f(A,B)=f(A,C)=f(B,C)=d$.
$a)$ Prove that $d$ is even.
$b)$ Prove that there exists a $n-$tuple $D=(d_1,\cdots,d_n)$ that $d_i=0 \ or \ 1$ for $i=1,2,\cdots,n$, such that $f(A,D)=f(B,D)=f(C,D)=\frac{d}{2}$.
1976 Canada National Olympiad, 8
Each of the 36 line segments joining 9 distinct points on a circle is coloured either red or blue. Suppose that each triangle determined by 3 of the 9 points contains at least one red side. Prove that there are four points such that the 6 segments connecting them are all red.
2001 Brazil National Olympiad, 6
A one-player game is played as follows: There is a bowl at each integer on the $Ox$-axis. All the bowls are initially empty, except for that at the origin, which contains $n \geq 2$ stones. A move is either
(A) to remove two stones from a bowl and place one in each of the two adjacent bowls, or
(B) to remove a stone from each of two adjacent bowls and to add one stone to the bowl immediately to their left.
Show that only a finite number of moves can be made and that the final position (when no more moves are possible) is independent of the moves made (for a given $n$).
1992 Turkey Team Selection Test, 3
A circle with radius $4$ and $251$ distinct points inside the circle are given. Show that it is possible to draw a circle with radius $1$ and containing at least $11$ of these points.
2013 Iran Team Selection Test, 7
Nonnegative real numbers $p_{1},\ldots,p_{n}$ and $q_{1},\ldots,q_{n}$ are such that $p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}$
Among all the matrices with nonnegative entries having $p_i$ as sum of the $i$-th row's entries and $q_j$ as sum of the $j$-th column's entries, find the maximum sum of the entries on the main diagonal.
2007 Macedonia National Olympiad, 5
Let $n$ be a natural number divisible by $4$. Determine the number of bijections $f$ on the set $\{1,2,...,n\}$ such that $f (j )+f^{-1}(j ) = n+1$ for $j = 1,..., n.$
1988 Canada National Olympiad, 3
Suppose that $S$ is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle
(i) whose vertices are all the same colour, and
(ii) at least one side of the triangle does not contain a point of the opposite colour.
2014 Indonesia MO Shortlist, C1
Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?
2010 Contests, 3
A total of $2010$ coins are distributed in $5$ boxes. At the beginning the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that she can do the following transformation finitely many times: from a box with at least 4 coins she can transfer one coin to each of the other boxes.
What is the maximum number of coins that Martha can take away?
2014 Miklós Schweitzer, 2
Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that
\[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \]
where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.
2008 Indonesia MO, 2
In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer.
2013 Bulgaria National Olympiad, 3
The integer lattice in the plane is colored with 3 colors. Find the least positive real $S$ with the property: for any such coloring it is possible to find a monochromatic lattice points $A,B,C$ with $S_{\triangle ABC}=S$.
[i]Proposed by Nikolay Beluhov[/i]
EDIT: It was the problem 3 (not 2), corrected the source title.
2012 Iran MO (3rd Round), 7
The city of Bridge Village has some highways. Highways are closed curves that have intersections with each other or themselves in $4$-way crossroads. Mr.Bridge Lover, mayor of the city, wants to build a bridge on each crossroad in order to decrease the number of accidents. He wants to build the bridges in such a way that in each highway, cars pass above a bridge and under a bridge alternately. By knowing the number of highways determine that this action is possible or not.
[i]Proposed by Erfan Salavati[/i]