Found problems: 1800
2014 IMS, 9
Let $G$ be a $2n-$vertices simple graph such that in any partition of the set of vertices of $G$ into two $n-$vertices sets $V_1$ and $V_2$, the number of edges from a vertex in $V_1$ to another vertex in $V_1$ is equal to the number of edges from a vertex in $V_2$ to another vertex in $V_2$. Prove that all the vertices have equal degrees.
2015 Romanian Master of Mathematics, 3
A finite list of rational numbers is written on a blackboard. In an [i]operation[/i], we choose any two numbers $a$, $b$, erase them, and write down one of the numbers \[
a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}.
\] Prove that, for every integer $n > 100$, there are only finitely many integers $k \ge 0$, such that, starting from the list \[ k + 1, \; k + 2, \; \dots, \; k + n, \] it is possible to obtain, after $n - 1$ operations, the value $n!$.
1999 Turkey Team Selection Test, 3
Prove that the plane is not a union of the inner regions of finitely many parabolas. (The outer region of a parabola is the union of the lines not intersecting the parabola. The inner region of a parabola is the set of points of the plane that do not belong to the outer region of the parabola)
2001 India IMO Training Camp, 3
Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$.
2011 Singapore MO Open, 2
If 46 squares are colored red in a $9\times 9$ board, show that there is a $2\times 2$ block on the board in which at least 3 of the squares are colored red.
2003 Tuymaada Olympiad, 3
Alphabet $A$ contains $n$ letters. $S$ is a set of words of finite length composed of letters of $A$. It is known that every infinite sequence of letters of $A$ begins with one and only one word of $S$.
Prove that the set $S$ is finite.
[i]Proposed by F. Bakharev[/i]
1997 Brazil National Olympiad, 2
Let $A$ be a set of $n$ non-negative integers. We say it has property $\mathcal P$ if the set $\{x + y \mid x, y \in A\}$ has $\binom{n}{2}$ elements. We call the largest element of $A$ minus the smallest element, the diameter of $A$. Let $f(n)$ be the smallest diameter of any set $A$ with property $\mathcal P$. Show that $n^2 \leq 4 f(n) < 4 n^3$.
[hide="Comment"](If you have some amount of time, try a best estimative for $f(n)$, such that $f(p)<2p^2$ for prime $p$).[/hide]
1990 USAMO, 4
Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left. (Your answer should be an explicit function of $n$ in simplest form.)
2019 Turkey Junior National Olympiad, 4
There are $27$ cardboard and $27$ plastic boxes. There are balls of certain colors inside the boxes. It is known that any two boxes of the same kind do not have a ball with the same color. Boxes of different kind have at least one ball of the same color. At each step we select two boxes that have a ball of same color and switch this common color into any other color we wish. Find the smallest number $n$ of moves required.
1991 Iran MO (2nd round), 3
Three groups $A, B$ and $C$ of mathematicians from different countries have invited to a ceremony. We have formed meetings such that three mathematicians participate in every meeting and there is exactly one mathematician from each group in every meeting. Also every two mathematicians have participated in exactly one meeting with each other.
[b](a)[/b] Prove that if this is possible, then number of mathematicians of the groups is equal.
[b](b)[/b] Prove that if there exist $3$ mathematicians in each group, then that work is possible.
[b](c)[/b] Prove that if number mathematicians of the groups be equal, then that work is possible.
2002 All-Russian Olympiad, 1
Can the cells of a $2002 \times 2002$ table be filled with the numbers from $1$ to $2002^2$ (one per cell) so that for any cell we can find three numbers $a, b, c$ in the same row or column (or the cell itself) with $a = bc$?
2011 Baltic Way, 6
Let $n$ be a positive integer. Prove that the number of lines which go through the origin and precisely one other point with integer coordinates $(x,y),0\le x,y\le n$, is at least $\frac{n^2}{4}$.
2011 Kosovo National Mathematical Olympiad, 5
Let $n>1$ be an integer and $S_n$ the set of all permutations $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$ where $\pi$ is bijective function. For every permutation $\pi \in S_n$ we define:
\[ F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) \]
where $M_n$ is taken with all permutations $\pi \in S_n$. Calculate the sum $M_n$.
2007 Baltic Way, 9
A society has to elect a board of governors. Each member of the society has chosen $10$ candidates for the board, but he will be happy if at least one of them will be on the board. For each six members of the society there exists a board consisting of two persons making all of these six members happy. Prove that a board consisting of $10$ persons can be elected making every member of the society happy.
2007 Tournament Of Towns, 2
A convex figure $F$ is such that any equilateral triangle with side $1$ has a parallel translation that takes all its vertices to the boundary of $F$. Is $F$ necessarily a circle?
2009 South East Mathematical Olympiad, 8
In an ${8}$×${8}$ squares chart , we dig out $n$ squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart .
Then find the mininum value of $n$ .
2014 Baltic Way, 7
Let $p_1, p_2, . . . , p_{30}$ be a permutation of the numbers $1, 2, . . . , 30.$ For how many permutations does the equality $\sum^{30}_{k=1}|p_k - k| = 450 $ hold?
2010 Macedonia National Olympiad, 5
Let the boxes in picture $1$ be marked as in picture $2$ below (from top to bottom in layers). In one move it is allowed to switch the empty box with another box adjacent to it (two boxes are adjacent if they share a common side). Can the arrangement of the numbers in picture $3$ be obtained after finitely many moves?
1992 Iran MO (2nd round), 3
There are some cities in both sides of a river and there are some sailing channels between the cities. Each sailing channel connects exactly one city from a side of the river to a city on the other side. Each city has exactly $k$ sailing channels. For every two cities, there's a way which connects them together. Prove that if we remove any (just one) sailing channel, then again for every two cities, there's a way that connect them together. $( k \geq 2)$
2005 Bulgaria Team Selection Test, 6
In a group of nine persons it is not possible to choose four persons such that every one knows the three others. Prove that this group of nine persons can be partitioned into four groups such that nobody knows anyone from his or her group.
2004 Romania National Olympiad, 4
Let $p,q \in \mathbb N^{\ast}$, $p,q \geq 2$. We say that a set $X$ has the property $\left( \mathcal S \right)$ if no matter how we choose $p$ subsets $B_i \subset X$, $i = \overline{1,n}$, not necessarily distinct, each with $q$ elements, there is a subset $Y \subset X$ with $p$ elements s.t. the intersection of $Y$ with each of the $B_i$'s has an element at most, $i=\overline{1,p}$. Prove that:
(a) if $p=4,q=3$ then any set composed of $9$ elements doesn't have $\left( \mathcal S \right)$;
(b) any set $X$ composed of $pq-q$ elements doesn't have the property $\left( \mathcal S \right)$;
(c) any set $X$ composed of $pq-q+1$ elements has the property $\left( \mathcal S \right)$.
[i]Dan Schwarz[/i]
2008 CentroAmerican, 4
Five girls have a little store that opens from Monday through Friday. Since two people are always enough for taking care of it, they decide to do a work plan for the week, specifying who will work each day, and fulfilling the following conditions:
a) Each girl will work exactly two days a week
b) The 5 assigned couples for the week must be different
In how many ways can the girls do the work plan?
2012 Turkey Junior National Olympiad, 4
We want to place $2012$ pockets, including variously colored balls, into $k$ boxes such that
[b]i)[/b] For any box, all pockets in this box must include a ball with the same color
or
[b]ii)[/b] For any box, all pockets in this box must include a ball having a color which is not included in any other pocket in this box
Find the smallest value of $k$ for which we can always do this placement whatever the number of balls in the pockets and whatever the colors of balls.
2013 Balkan MO, 4
In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$.
We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle.
The following property is satisfied:
"for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element"
Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends.
([i]Serbia[/i])
2012 Macedonia National Olympiad, 5
A hexagonal table is given, as the one on the drawing, which has $~$ $2012$ $~$ columns. There are $~$ $2012$ $~$ hexagons in each of the odd columns, and there are $~$ $2013$ $~$ hexagons in each of the even columns. The number $~$ $i$ $~$ is written in each hexagon from the $~$ $i$-th column. Changing the numbers in the table is allowed in the following way: We arbitrarily select three adjacent hexagons, we rotate the numbers, and if the rotation is clockwise then the three numbers decrease by one, and if we rotate them counterclockwise the three numbers increase by one (see the drawing below). What's the maximum number of zeros that can be obtained in the table by using the above-defined steps.