Found problems: 1800
2014 Abels Math Contest (Norwegian MO) Final, 3a
A grasshopper is jumping about in a grid. From the point with coordinates $(a, b)$ it can jump to either $(a + 1, b),(a + 2, b),(a + 1, b + 1),(a, b + 2)$ or $(a, b + 1)$. In how many ways can it reach the line $x + y = 2014?$ Where the grasshopper starts in $(0, 0)$.
2014 Contests, 3
Let $n$ a positive integer. In a $2n\times 2n$ board, $1\times n$ and $n\times 1$ pieces are arranged without overlap.
Call an arrangement [b]maximal[/b] if it is impossible to put a new piece in the board without overlapping the previous ones.
Find the least $k$ such that there is a [b]maximal[/b] arrangement that uses $k$ pieces.
2014 Baltic Way, 9
What is the least posssible number of cells that can be marked on an $n \times n$ board such that for each $m >\frac{ n}{2}$ both diagonals of any $m \times m$ sub-board contain a marked cell?
2008 China Team Selection Test, 3
Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\leq i < j < k\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle.
2005 Baltic Way, 9
A rectangle is divided into $200\times 3$ unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size $1\times 2$ is divisible by $3$.
1982 IMO Longlists, 52
We are given $2n$ natural numbers
\[1, 1, 2, 2, 3, 3, \ldots, n - 1, n - 1, n, n.\]
Find all $n$ for which these numbers can be arranged in a row such that for each $k \leq n$, there are exactly $k$ numbers between the two numbers $k$.
2013 Stars Of Mathematics, 4
A set $S$ of unit cells of an $n\times n$ array, $n\geq 2$, is said [i]full[/i] if each row and each column of the array contain at least one element of $S$, but which has this property no more when any of its elements is removed. A full set having maximum cardinality is said [i]fat[/i], while a full set of minimum cardinality is said [i]meagre[/i].
i) Determine the cardinality $m(n)$ of the meagre sets, describe all meagre sets and give their count.
ii) Determine the cardinality $M(n)$ of the fat sets, describe all fat sets and give their count.
[i](Dan Schwarz)[/i]
2011 All-Russian Olympiad Regional Round, 9.8
Straight rod of 2 meter length is cut into $N$ sticks. The length of each piece is an integer number of centimeters. For which smallest $N$ can one guarantee that it is possible to form the contour of some rectangle, while using all sticks and not breaking them further?
(Author: A. Magazinov)
2004 Turkey Team Selection Test, 3
Each student in a classroom has $0,1,2,3,4,5$ or $6$ pieces of candy. At each step the teacher chooses some of the students, and gives one piece of candy to each of them and also to any other student in the classroom who is friends with at least one of these students. A student who receives the seventh piece eats all $7$ pieces. Assume that for every pair of students in the classroom, there is at least one student who is friend swith exactly one of them. Show that no matter how many pieces each student has at the beginning, the teacher can make them to have any desired numbers of pieces after finitely many steps.
2005 Taiwan TST Round 3, 3
The set $\{1,2,\dots\>,n\}$ is called $P$. The function $f: P \to \{1,2,\dots\>,m\}$ satisfies \[f(A\cap B)=\min (f(A), f(B)).\] What is the relationship between the number of possible functions $f$ with the sum $\displaystyle \sum_{j=1}^m j^n$?
There is a nice and easy solution to this. Too bad I did not think of it...
2013 China Team Selection Test, 1
For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$.
Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.
1974 IMO Longlists, 38
The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of:
[b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes;
[b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes;
[b](c)[/b] squares in total, with vertices on the lattice.
2012 Romanian Masters In Mathematics, 1
Given a finite number of boys and girls, a [i]sociable set of boys[/i] is a set of boys such that every girl knows at least one boy in that set; and a [i]sociable set of girls[/i] is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.)
[i](Poland) Marek Cygan[/i]
2014 China Team Selection Test, 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 )
2002 Iran MO (3rd Round), 13
$f,g$ are two permutations of set $X=\{1,\dots,n\}$. We say $f,g$ have common points iff there is a $k\in X$ that $f(k)=g(k)$.
a) If $m>\frac{n}{2}$, prove that there are $m$ permutations $f_{1},f_{2},\dots,f_{m}$ from $X$ that for each permutation $f\in X$, there is an index $i$ that $f,f_{i}$ have common points.
b) Prove that if $m\leq\frac{n}{2}$, we can not find permutations $f_{1},f_{2},\dots,f_{m}$ satisfying the above condition.
1971 IMO Longlists, 33
A square $2n\times 2n$ grid is given. Let us consider all possible paths along grid lines, going from the centre of the grid to the border, such that (1) no point of the grid is reached more than once, and (2) each of the squares homothetic to the grid having its centre at the grid centre is passed through only once.
(a) Prove that the number of all such paths is equal to $4\prod_{i=2}^n(16i-9)$.
(b) Find the number of pairs of such paths that divide the grid into two congruent figures.
(c) How many quadruples of such paths are there that divide the grid into four congruent parts?
2015 China Team Selection Test, 5
Set $S$ to be a subset of size $68$ of $\{1,2,...,2015\}$. Prove that there exist $3$ pairwise disjoint, non-empty subsets $A,B,C$ such that $|A|=|B|=|C|$ and $\sum_{a\in A}a=\sum_{b\in B}b=\sum_{c\in C}c$
2009 Kazakhstan National Olympiad, 3
In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game.
If after ending of tournament participant have at least $ 75 %
$ of maximum possible points he called $winner$ $of$ $tournament$.
Find maximum possible numbers of $winners$ $of$ $tournament$.
2004 Italy TST, 2
Let $\mathcal{P}_0=A_0A_1\ldots A_{n-1}$ be a convex polygon such that $A_iA_{i+1}=2^{[i/2]}$ for $i=0, 1,\ldots ,n-1$ (where $A_n=A_0$). Define the sequence of polygons $\mathcal{P}_k=A_0^kA_1^k\ldots A_{n-1}^k$ as follows: $A_i^1$ is symmetric to $A_i$ with respect to $A_0$, $A_i^2$ is symmetric to $A_i^1$ with respect to $A_1^1$, $A_i^3$ is symmetric to $A_i^2$ with respect to $A_2^2$ and so on. Find the values of $n$ for which infinitely many polygons $\mathcal{P}_k$ coincide with $\mathcal{P}_0$.
2011 IberoAmerican, 3
Let $k$ and $n$ be positive integers, with $k \geq 2$. In a straight line there are $kn$ stones of $k$ colours, such that there are $n$ stones of each colour. A [i]step[/i] consists of exchanging the position of two adjacent stones. Find the smallest positive integer $m$ such that it is always possible to achieve, with at most $m$ steps, that the $n$ stones are together, if:
a) $n$ is even.
b) $n$ is odd and $k=3$
2012 Gulf Math Olympiad, 3
Consider a $3\times7$ grid of squares. Each square may be coloured green or white.
[list]
(a) Is it possible to find a colouring so that no subrectangle has all four corner squares of the same colour?
(b) Is it possible for a $4\times 6$ grid?
[/list]
[i]Subrectangles must have their corners at grid-points of the original diagram. The corner squares of a subrectangle must be different. The original diagram is a subrectangle of itself.[/i]
2001 Iran MO (2nd round), 3
Find all positive integers $n$ such that we can put $n$ equal squares on the plane that their sides are horizontal and vertical and the shape after putting the squares has at least $3$ axises.
2009 Portugal MO, 3
Duarte wants to draw a square whose side's length is $2009$ cm and which is divided in $2009\times2009$ squares whose side's length is $1$ cm and whose sides are parallel to the original square's one, without taking the pencil out of the paper. Starting on one of the vertex of the giant square, what is the length of the shortest line that allows him to make this drawing?
2010 Indonesia TST, 4
Prove that the number $ (\underbrace{9999 \dots 99}_{2005}) ^{2009}$ can be obtained by erasing some digits of $ (\underbrace{9999 \dots 99}_{2008}) ^{2009}$ (both in decimal representation).
[i]Yudi Satria, Jakarta[/i]
2011 Iran MO (2nd Round), 2
rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ and $l$ different from the colors it has in days $i$ and $k$. what is the maximum number of days rainbow can live in terms of $n$?