Found problems: 1488
1984 IMO Longlists, 23
A $2\times 2\times 12$ box fixed in space is to be filled with twenty-four $1 \times 1 \times 2$ bricks. In how many ways can this be done?
2014 Spain Mathematical Olympiad, 1
Is it possible to place the numbers $0,1,2,\dots,9$ on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?
2007 India IMO Training Camp, 3
Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define
\[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\]
Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$
(Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)
1992 China National Olympiad, 3
Given a $9\times 9$ grid, we assign either $+1$ or $-1$ to each square on the grid. We define an [i]adjustment[/i] as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have $81$ products. Next we replace all the squares’ values with their corresponding products. Determine if we can make all values in the grid equal to $1$ through finite [i]adjustments[/i].
2010 India IMO Training Camp, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
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.
2012 Balkan MO, 3
Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$
Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$
2004 All-Russian Olympiad, 2
A country has 1001 cities, and each two cities are connected by a one-way street. From each city exactly 500 roads begin, and in each city 500 roads end. Now an independent republic splits itself off the country, which contains 668 of the 1001 cities. Prove that one can reach every other city of the republic from each city of this republic without being forced to leave the republic.
2017 Bundeswettbewerb Mathematik, 2
What is the maximum number of acute interior angles a non-overlapping planar $2017$-gon can have?
2008 Costa Rica - Final Round, 1
We want to colour all the squares of an $ nxn$ board of red or black. The colorations should be such that any subsquare of $ 2x2$ of the board have exactly two squares of each color. If $ n\geq 2$ how many such colorations are possible?
2014 Saudi Arabia IMO TST, 2
Define a [i]domino[/i] to be an ordered pair of [i]distinct[/i] positive integers. A [i]proper sequence[/i] of dominoes is a list of distinct dominoes in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i, j)$ and $(j, i)$ do not [i]both[/i] appear for any $i$ and $j$. Let $D_n$ be the set of all dominoes whose coordinates are no larger than $n$. Find the length of the longest proper sequence of dominoes that can be formed using the dominoes of $D_n$.
2006 Bulgaria Team Selection Test, 3
[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$?
[i]Alexandar Ivanov[/i]
2012 USA Team Selection Test, 4
There are 2010 students and 100 classrooms in the Olympiad High School. At the beginning, each of the students is in one of the classrooms. Each minute, as long as not everyone is in the same classroom, somebody walks from one classroom into a different classroom with at least as many students in it (prior to his move). This process will terminate in $M$ minutes. Determine the maximum value of $M$.
2003 Bulgaria National Olympiad, 1
A set $A$ of positive integers is called [i]uniform[/i] if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer $n>1$ such that there exist a uniform set $A$ with $n$ elements.
1984 IMO Longlists, 34
One country has $n$ cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number of flights that the worker will have to use?
1990 China Team Selection Test, 1
In a wagon, every $m \geq 3$ people have exactly one common friend. (When $A$ is $B$'s friend, $B$ is also $A$'s friend. No one was considered as his own friend.) Find the number of friends of the person who has the most friends.
2006 France Team Selection Test, 3
Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$
Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$
[i]Edited by orl.[/i]
1993 All-Russian Olympiad Regional Round, 11.4
Given a regular $ 2n$-gon, show that each of its sides and diagonals can be assigned in such a way that the sum of the obtained vectors equals zero.
1985 IMO Longlists, 91
Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of people they had shaken hands with, and all the answers she got were different. How many people did the deputy leader of the Illyrian team greet ?
2003 Tournament Of Towns, 6
The signs "$+$" or "$-$" are placed in all cells of a $4 \times 4$ square table. It is allowed to change a sign of any cell altogether with signs of all its adjacent cells (i.e. cells having a common side with it). Find the number of different tables that could be obtained by iterating this procedure.
2002 All-Russian Olympiad, 2
Several points are given in the plane. Suppose that for any three of them, there exists an orthogonal coordinate system (determined by the two axes and the unit length) in which these three points have integer coordinates. Prove that there exists an orthogonal coordinate system in which all the given points have integer coordinates.
2011 Czech-Polish-Slovak Match, 2
Written on a blackboard are $n$ nonnegative integers whose greatest common divisor is $1$. A [i]move[/i] consists of erasing two numbers $x$ and $y$, where $x\ge y$, on the blackboard and replacing them with the numbers $x-y$ and $2y$. Determine for which original $n$-tuples of numbers on the blackboard is it possible to reach a point, after some number of moves, where $n-1$ of the numbers of the blackboard are zeroes.
2008 Hong Kong TST, 1
In a school there are $ 2008$ students. Students are members of certain committees. A committee has at most $ 1004$ members and every two students join a common committee.
(i) Determine the smallest possible number of committees in the school.
(ii) If it is further required that the union of any two committees consists of at most $ 1800$ students, will your answer in (i) still hold?
2014 Postal Coaching, 2
Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.
2014 Iran Team Selection Test, 6
Consider $n$ segments in the plane which no two intersect and between their $2n$ endpoints no three are collinear. Is the following statement true?
Statement: There exists a simple $2n$-gon such that it's vertices are the $2n$ endpoints of the segments and each segment is either completely inside the polygon or an edge of the polygon.