Found problems: 1800
2013 Serbia National Math Olympiad, 4
Determine all natural numbers $n$ for which there is a partition of $\{1,2,...,3n\}$ in $n$ pairwise disjoint subsets of the form $\{a,b,c\}$, such that numbers $b-a$ and $c-b$ are different numbers from the set $\{n-1, n, n+1\}$.
2024 Baltic Way, 7
A $45 \times 45$ grid has had the central unit square removed. For which positive integers $n$ is it possible to cut the remaining area into $1 \times n$ and $n\times 1$ rectangles?
1999 Baltic Way, 9
A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1, 2,\ldots ,27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the cube). At most how many of the $27$ row sums can be odd?
1996 All-Russian Olympiad, 4
In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members.
[i]A. Skopenkov[/i]
2004 Pre-Preparation Course Examination, 7
Let $ G=(V,E)$ be a simple graph.
a) Let $ A,B$ be a subsets of $ E$, and spanning subgraphs of $ G$ with edges $ A,B,A\cup B$ and $ A\cap B$ have $ a,b,c$ and $ d$ connected components respectively. Prove that $ a+b\leq c+d$.
We say that subsets $ A_1,A_2,\dots,A_m$ of $ E$ have $ (R)$ property if and only if for each $ I\subset\{1,2,\dots,m\}$ the spanning subgraph of $ G$ with edges $ \cup_{i\in I}A_i$ has at most $ n-|I|$ connected components.
b) Prove that when $ A_1,\dots,A_m,B$ have $ (R)$ property, and $ |B|\geq2$, there exists an $ x\in B$ such that $ A_1,A_2,\dots,A_m,B\backslash\{x\}$ also have property $ (R)$.
Suppose that edges of $ G$ are colored arbitrarily. A spanning subtree in $ G$ is called colorful if and only if it does not have any two edges with the same color.
c) Prove that $ G$ has a colorful subtree if and only if for each partition of $ V$ to $ k$ non-empty subsets such as $ V_1,\dots,V_k$, there are at least $ k\minus{}1$ edges with distinct colors that each of these edges has its two ends in two different $ V_i$s.
d) Assume that edges of $ K_n$ has been colored such that each color is repeated $ \left[\frac n2\right]$ times. Prove that there exists a colorful subtree.
e) Prove that in part d) if $ n\geq5$ there is a colorful subtree that is non-isomorphic to $ K_{1,n-1}$.
f) Prove that in part e) there are at least two non-intersecting colorful subtrees.
2013 National Olympiad First Round, 36
A chess club consists of at least $10$ and at most $50$ members, where $G$ of them are female, and $B$ of them are male with $G>B$. In a chess tournament, each member plays with any other member exactly one time. At each game, the winner gains $1$, the loser gains $0$ and both player gains $1/2$ point when a tie occurs. At the tournament, it is observed that each member gained exactly half of his/her points from the games played against male members. How many different values can $B$ take?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 1
$
1999 Brazil National Olympiad, 3
How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?
2008 Danube Mathematical Competition, 4
Let $ n\geq 2$ be a positive integer. Find the [u]maximum[/u] number of segments with lenghts greater than $ 1,$ determined by $ n$ points which lie on a closed disc with radius $ 1.$
2012 Turkey Team Selection Test, 1
Let $A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}$ and $S$ be the set of all subsets of $A.$ Find the number of functions $f : S\to B$ satisfying $f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}$ for all $A_1, A_2 \in S.$
2011 China Second Round Olympiad, 3
Given $n\ge 4$ real numbers $a_{n}>...>a_{1} > 0$. For $r > 0$, let $f_{n}(r)$ be the number of triples $(i,j,k)$ with $1\leq i<j<k\leq n$ such that $\frac{a_{j}-a_{i}}{a_{k}-a_{j}}=r$. Prove that ${f_{n}(r)}<\frac{n^{2}}{4}$.
2012 Serbia National Math Olympiad, 2
Let $\mathbb{K}$ be two-dimensional integer lattice. Is there a bijection $f:\mathbb{N} \rightarrow \mathbb{K}$, such that for every distinct $a,b,c \in \mathbb{N}$ we have: \[\gcd(a,b,c)>1 \Rightarrow f(a),f(b),f(c) \mbox{ are not colinear? }\]
1977 IMO Longlists, 39
Consider $37$ distinct points in space, all with integer coordinates. Prove that we may find among them three distinct points such that their barycentre has integers coordinates.
2013 European Mathematical Cup, 3
We are given a combination lock consisting of $6$ rotating discs. Each disc consists of digits $0, 1, 2,\ldots , 9$ in that order (after digit $9$ comes $0$). Lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction and the lock opens instantly if the current combination is correct. Discs are initially put in the position $000000$, and we know that this combination is not correct.
[list]
a) What is the least number of moves necessary to ensure that we have found the correct combination?
b) What is the least number of moves necessary to ensure that we have found the correct combination, if we
know that none of the combinations $000000, 111111, 222222, \ldots , 999999$ is correct?[/list]
[i]Proposed by Ognjen Stipetić and Grgur Valentić[/i]
2013 Turkey Junior National Olympiad, 4
Player $A$ places an odd number of boxes around a circle and distributes $2013$ balls into some of these boxes. Then the player $B$ chooses one of these boxes and takes the balls in it. After that the player $A$ chooses half of the remaining boxes such that none of two are consecutive and take the balls in them. If player $A$ guarantees to take $k$ balls, find the maximum possible value of $k$.
2012 Brazil Team Selection Test, 2
Into each box of a $ 2012 \times 2012 $ square grid, a real number greater than or equal to $ 0 $ and less than or equal to $ 1 $ is inserted. Consider splitting the grid into $2$ non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to $ 1 $, no matter how the grid is split into $2$ such rectangles. Determine the maximum possible value for the sum of all the $ 2012 \times 2012 $ numbers inserted into the boxes.
2004 Iran MO (2nd round), 3
The road ministry has assigned $80$ informal companies to repair $2400$ roads. These roads connect $100$ cities to each other. Each road is between $2$ cities and there is at most $1$ road between every $2$ cities. We know that each company repairs $30$ roads that it has agencies in each $2$ ends of them. Prove that there exists a city in which $8$ companies have agencies.
2011 ITAMO, 6
Let $X = \{1, 2, 3, 4, 5, 6, 7, 8\}$. We want to color, using $k$ colors, all subsets of $3$ elements of $X$ in such a way that, two disjoint subsets have distinct colors.
Prove that:
(a) $4$ colors are sufficient;
(b) $3$ colors are not sufficient.
2007 Italy TST, 1
We have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?
2008 China National Olympiad, 2
Given an integer $n\ge3$, prove that the set $X=\{1,2,3,\ldots,n^2-n\}$ can be divided into two non-intersecting subsets such that neither of them contains $n$ elements $a_1,a_2,\ldots,a_n$ with $a_1<a_2<\ldots<a_n$ and $a_k\le\frac{a_{k-1}+a_{k+1}}2$ for all $k=2,\ldots,n-1$.
2007 Kyiv Mathematical Festival, 5
a) One has a set of stones with weights $1, 2, \ldots, 20$ grams. Find all $k$ for which it is possible to place $k$ and the rest $20-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved.
b) One has a set of stones with weights $1, 2, \ldots, 51$ grams. Find all $k$ for which it is possible to place $k$ and the rest $51-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved.
c) One has a set of stones with weights $1, 2, \ldots, n$ grams ($n\in\mathbb{N}$). Find all $n$ and $k$ for which it is possible to place $k$ and the rest $n-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved.
[size=75] a) and b) were proposed at the festival, c) is a generalization[/size]
2006 Moldova National Olympiad, 11.4
On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.
2014 Contests, 2
Let $A$ be a finite set of positive numbers , $B=\{\frac{a+b}{c+d} |a,b,c,d \in A \}$.
Show that: $\left | B \right | \ge 2\left | A \right |^2-1 $,
where $|X| $ be the number of elements of the finite set $X$.
(High School Affiliated to Nanjing Normal University )
2014 National Olympiad First Round, 28
The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. How many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
2012 All-Russian Olympiad, 3
A plane is coloured into black and white squares in a chessboard pattern. Then, all the white squares are coloured red and blue such that any two initially white squares that share a corner are different colours. (One is red and the other is blue.) Let $\ell$ be a line not parallel to the sides of any squares. For every line segment $I$ that is parallel to $\ell$, we can count the difference between the length of its red and its blue areas. Prove that for every such line $\ell$ there exists a number $C$ that exceeds all those differences that we can calculate.
2004 Italy TST, 3
Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.