Found problems: 1488
2004 Austrian-Polish Competition, 8
a.) Prove that for $n = 4$ or $n \geq 6$ each triangle $ABC$ can be decomposed in $n$ similar (not necessarily congruent) triangles.
b.) Show: An equilateral triangle can neither be composed in 3 nor 5 triangles.
c.) Is there a triangle $ABC$ which can be decomposed in 3 and 5 triangles, analogously to a.). Either give an example or prove that there is not such a triangle.
2001 South africa National Olympiad, 4
$n$ red and $n$ blue points on a plane are given so that no three of the $2n$ points are collinear. Prove that it is always possible to split up the points into $n$ pairs, with one red and one blue point in each pair, so that no two of the $n$ line segments which connect the two members of a pair intersect.
2011 India IMO Training Camp, 3
Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.
1984 IMO Longlists, 57
Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8,4$ and let $n = (10a + b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n.$
1998 Brazil Team Selection Test, Problem 1
Let N be a positive integer greater than 2. We number the vertices
of a regular 2n-gon clockwise with the numbers 1, 2, . . . ,N,−N,−N +
1, . . . ,−2,−1. Then we proceed to mark the vertices in the following way.
In the first step we mark the vertex 1. If ni is the vertex marked in the
i-th step, in the i+1-th step we mark the vertex that is |ni| vertices away
from vertex ni, counting clockwise if ni is positive and counter-clockwise
if ni is negative. This procedure is repeated till we reach a vertex that has
already been marked. Let $f(N)$ be the number of non-marked vertices.
(a) If $f(N) = 0$, prove that 2N + 1 is a prime number.
(b) Compute $f(1997)$.
1996 Polish MO Finals, 3
From the set of all permutations $f$ of $\{1, 2, ... , n\}$ that satisfy the condition:
$f(i) \geq i-1$ $i=1,...,n$
one is chosen uniformly at random. Let $p_n$ be the probability that the chosen permutation $f$ satisfies
$f(i) \leq i+1$ $i=1,...,n$
Find all natural numbers $n$ such that $p_n > \frac{1}{3}$.
2007 Croatia Team Selection Test, 4
Given a finite string $S$ of symbols $X$ and $O$, we write $@(S)$ for the number of $X$'s in $S$ minus the number of $O$'s. (For example, $@(XOOXOOX) =-1$.) We call a string $S$ [b]balanced[/b] if every substring $T$ of (consecutive symbols) $S$ has the property $-2 \leq @(T) \leq 2$. (Thus $XOOXOOX$ is not balanced since it contains the sub-string $OOXOO$ whose $@$-value is $-3$.) Find, with proof, the number of balanced strings of length $n$.
1995 Vietnam Team Selection Test, 1
A graph has $ n$ vertices and $ \frac {1}{2}\left(n^2 \minus{} 3n \plus{} 4\right)$ edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length $ k$ of a circuit in such a graph.
2006 Estonia Math Open Senior Contests, 2
After the schoolday is over, Juku must attend an extra math class. The teacher
writes a quadratic equation $ x^2\plus{} p_1x\plus{}q_1 \equal{} 0$ with integer coefficients on the blackboard and Juku has to find its solutions. If they are not both integers, Jukumay go home. If the solutions are integers, then the teacher writes a new equation $ x^2 \plus{} p_2x \plus{} q_2 \equal{} 0,$ where $ p_2$ and $ q_2$ are the solutions of the previous equation taken in some order, and everything starts all over. Find all possible values for $ p_1$ and $ q_1$ such that the teacher can hold Juku at school forever.
2013 Czech-Polish-Slovak Match, 2
Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle.
[asy]
unitsize(0.25cm);
path p=polygon(3);
for(int m=0; m<=11;++m){
for(int n=0 ; n<= 11-m; ++n){
draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p);
}
}
[/asy]
2008 Argentina Iberoamerican TST, 3
The plane is divided into regions by $ n \ge 3$ lines, no two of which are parallel, and no three of which are concurrent. Some regions are coloured , in such a way that no two coloured regions share a common segment or half-line of their borders. Prove that the number of coloured regions is at most $ \frac{n(n\plus{}1)}{3}$
2010 China National Olympiad, 2
There is a deck of cards placed at every points $A_1, A_2, \ldots , A_n$ and $O$, where $n \geq 3$. We can do one of the following two operations at each step:
$1)$ If there are more than 2 cards at some points $A_i$, we can withdraw three cards from that deck and place one each at $A_{i-1}, A_{i+1}$ and $O$. (Here $A_0=A_n$ and $A_{n+1}=A_1$);
$2)$ If there are more than or equal to $n$ cards at point $O$, we can withdraw $n$ cards from that deck and place one each at $A_1, A_2, \ldots , A_n$.
Show that if the total number of cards is more than or equal to $n^2+3n+1$, we can make the number of cards at every points more than or equal to $n+1$ after finitely many steps.
2010 Tournament Of Towns, 4
At the math contest each participant met at least $3$ pals who he/she already knew. Prove that the Jury can choose an even number of participants (more than two) and arrange them around a table so that each participant be set between these who he/she knows.
2005 Vietnam Team Selection Test, 2
Given $n$ chairs around a circle which are marked with numbers from 1 to $n$ .There are $k$, $k \leq 4 \cdot n$ students sitting on those chairs .Two students are called neighbours if there is no student sitting between them. Between two neighbours students ,there are at less 3 chairs. Find the number of choices of $k$ chairs so that $k$ students can sit on those and the condition is satisfied.
2007 Olympic Revenge, 4
Let $A_{1}A_{2}B_{1}B_{2}$ be a convex quadrilateral. At adjacent vertices $A_{1}$ and $A_{2}$ there are two Argentinian cities. At adjacent vertices $B_{1}$ and $B_{2}$ there are two Brazilian cities. There are $a$ Argentinian cities and $b$ Brazilian cities in the quadrilateral interior, no three of which collinear. Determine if it's possible, independently from the cities position, to build straight roads, each of which connects two Argentinian cities ou two Brazilian cities, such that:
$\bullet$ Two roads does not intersect in a point which is not a city;
$\bullet$ It's possible to reach any Argentinian city from any Argentinian city using the roads; and
$\bullet$ It's possible to reach any Brazilian city from any Brazilian city using the roads.
If it's always possible, construct an algorithm that builds a possible set of roads.
2000 India National Olympiad, 6
For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that
(i) $f(1999) > f (1996)$;
(ii) $f(2000) = f(1997)$.
2002 Canada National Olympiad, 1
Let $S$ be a subset of $\{1, 2, \dots, 9\}$, such that the sums formed by adding each unordered pair of distinct numbers from $S$ are all different. For example, the subset $\{1, 2, 3, 5\}$ has this property, but $\{1, 2, 3, 4, 5\}$ does not, since the pairs $\{1, 4\}$ and $\{2, 3\}$ have the same sum, namely 5.
What is the maximum number of elements that $S$ can contain?
2004 Tournament Of Towns, 6
Two persons are dividing a piece of cheese. The first person cuts it into two pieces, then the second person cuts one of these pieces into two, then again the first person cuts one of the pieces into two, and so until they have 5 pieces. After that the first person chooses one of the pieces, then the second person chooses one of remaining pieces and so on until all pieces are taken. For each of the players, what is the maximal amount of cheese he can get for certain, regardless of the other's actions?
2005 Finnish National High School Mathematics Competition, 5
A finite sequence is said to be [i]disorderly[/i], if no two terms of the sequence have their average in between them. For example, $(0, 2, 1)$ is disorderly, for $1 = \frac{0+2}{2}$ is not in between $0$ and $2$, and the other averages $\frac{0+1}{2} = \frac{1}{2}$ and $\frac{2+1}{2} = 1\frac{1}{2}$ do not even occur in the sequence.
Prove that for every $n \in \Bbb{N}$ there is a disorderly sequence enumerating the numbers $0, 1,\ldots , n$ without repetitions.
1986 Vietnam National Olympiad, 3
A sequence of positive integers is constructed as follows: the first term is $ 1$, the following two terms are $ 2$, $ 4$, the following three terms are $ 5$, $ 7$, $ 9$, the following four terms are $ 10$, $ 12$, $ 14$, $ 16$, etc. Find the $ n$-th term of the sequence.
1988 China Team Selection Test, 4
Let $k \in \mathbb{N},$ $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$ Any two elements $(a, b)$, $(c, d)$ $\in S_k$ are called "undistinguishing" in $S_k$ if $a - c \equiv 0$ or $\pm 1 \pmod{k}$ and $b - d \equiv 0$ or $\pm 1 \pmod{k}$; otherwise, we call them "distinguishing". For example, $(1, 1)$ and $(2, 5)$ are undistinguishing in $S_5$. Considering the subset $A$ of $S_k$ such that the elements of $A$ are pairwise distinguishing. Let $r_k$ be the maximum possible number of elements of $A$.
(i) Find $r_5$.
(ii) Find $r_7$.
(iii) Find $r_k$ for $k \in \mathbb{N}$.
2012 Korea - Final Round, 3
$ A_1 , A_2 , \cdots , A_n $ are given subsets. Let $ S = \left\{ 1, 2, \cdots , n \right\} $. For any $ X \subset S $, let
\[ N(X)= \left\{ i \in S-X \ | \ \forall j \in X, \ A_i \cap A_j \ne \emptyset \right\} \]
Let $ m $ be an integer such that $ 3 \le m \le n-2 $. Prove that there exist $ X \subset S $ such that $ |X|=m $ and $ |N(X)| \ne 1 $.
2002 Brazil National Olympiad, 6
Show that we cannot form more than $4096$ binary sequences of length $24$ so that any two differ in at least $8$ positions.
2006 MOP Homework, 7
Let $A_{n,k}$ denote the set of lattice paths in the coordinate plane of upsteps $u=[1,1]$, downsteps $d=[1,-1]$, and flatsteps $f=[1,0]$ that contain $n$ steps, $k$ of which are slanted ($u$ or $d$). A sharp turn is a consecutive pair of $ud$ or $du$. Let $B_{n,k}$ denote the set of paths in $A_{n,k}$ with no upsteps among the first $k-1$ steps, and let $C_{n,k}$ denote the set of paths in $A_{n,k}$ with no sharps anywhere. For example, $fdu$ is in $B_{3,2}$ but not in $C_{3,2}$, while $ufd$ is in $C_{3,2}$ but not $B_{3,2}$. For $1 \le k \le n$, prove that the sets $B_{n,k}$ and $C_{n,k}$ contains the same number of elements.
1994 IberoAmerican, 3
In each square of an $n\times{n}$ grid there is a lamp. If the lamp is touched it changes its state every lamp in the same row and every lamp in the same column (the one that are on are turned off and viceversa). At the begin, all the lamps are off. Show that lways is possible, with an appropriated sequence of touches, that all the the lamps on the board end on and find, in function of $n$ the minimal number of touches that are necessary to turn on every lamp.