Found problems: 1800
2012 Iran MO (3rd Round), 6
[b]a)[/b] Prove that $a>0$ exists such that for each natural number $n$, there exists a convex $n$-gon $P$ in plane with lattice points as vertices such that the area of $P$ is less than $an^3$.
[b]b)[/b] Prove that there exists $b>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $bn^2$.
[b]c)[/b] Prove that there exist $\alpha,c>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $cn^{2+\alpha}$.
[i]Proposed by Mostafa Eynollahzade[/i]
1998 Spain Mathematical Olympiad, 3
Determine the values of $n$ for which an $n\times n$ square can be tiled with pieces of the type [img]http://oi53.tinypic.com/v3pqoh.jpg[/img].
2006 Pre-Preparation Course Examination, 7
Suppose that for every $n$ the number $m(n)$ is chosen such that $m(n)\ln(m(n))=n-\frac 12$. Show that $b_n$ is asymptotic to the following expression where $b_n$ is the $n-$th Bell number, that is the number of ways to partition $\{1,2,\ldots,n\}$:
\[ \frac{m(n)^ne^{m(n)-n-\frac 12}}{\sqrt{\ln n}}. \]
Two functions $f(n)$ and $g(n)$ are asymptotic to each other if $\lim_{n\rightarrow \infty}\frac{f(n)}{g(n)}=1$.
2013 India IMO Training Camp, 3
A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the following game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers $a, b$, neither of which was chosen earlier by any player and move the marker by $a$ units in the horizontal direction and $b$ units in the vertical direction. Hobbes wins if the marker is back at the origin any time after the first turn. Prove or disprove that Calvin can prevent Hobbes from winning.
Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).
2002 JBMO ShortLists, 1
A student is playing computer. Computer shows randomly 2002 positive numbers. Game's rules let do the following operations
- to take 2 numbers from these, to double first one, to add the second one and to save the sum.
- to take another 2 numbers from the remainder numbers, to double the first one, to add the second one, to multiply this sum with previous and to save the result.
- to repeat this procedure, until all the 2002 numbers won't be used.
Student wins the game if final product is maximum possible.
Find the winning strategy and prove it.
2019 Turkey EGMO TST, 6
There are $k$ piles and there are $2019$ stones totally. In every move we split a pile into two or remove one pile. Using finite moves we can reach conclusion that there are $k$ piles left and all of them contain different number of stonws. Find the maximum of $k$.
2002 ITAMO, 6
We are given a chessboard with 100 rows and 100 columns. Two squares of the board are said to be adjacent if they have a common side. Initially all squares are white.
a) Is it possible to colour an odd number of squares in such a way that each coloured square has an odd number of adjacent coloured squares?
b) Is it possible to colour some squares in such a way that an odd number of them have exactly $4$ adjacent coloured squares and all the remaining coloured squares have exactly $2$ adjacent coloured squares?
c) Is it possible to colour some squares in such a way that an odd number of them have exactly $2$ adjacent coloured squares and all the remaining coloured squares have exactly $4$ adjacent coloured squares?
2002 Tuymaada Olympiad, 1
Each of the points $G$ and $H$ lying from different sides of the plane of hexagon $ABCDEF$ is connected with all vertices of the hexagon.
Is it possible to mark 18 segments thus formed by the numbers $1, 2, 3, \ldots, 18$ and arrange some real numbers at points $A, B, C, D, E, F, G, H$ so that each segment is marked with the difference of the numbers at its ends?
[i]Proposed by A. Golovanov[/i]
2007 Pre-Preparation Course Examination, 2
There is a WORD game with the following rules. There are finite number of relations $U_{i}\longrightarrow V_{i}$($U_{i},V_{i}$ are words). There is are two words $A,B$. We start from $A$, and we want to reach to $B$. At each step we can change one subword $U_{i}$ to $V_{i}$. Prove that there does not exist an algorithm that picks up $A,B$ and $U_{i}$'s,$V_{i}$'s and decides whether we can reach from $A$ to $B$ or not.
2007 Romania Team Selection Test, 1
Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find
\[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]
1996 Baltic Way, 16
On an infinite checkerboard two players alternately mark one unmarked cell. One of them uses $\times$, the other $\circ$. The first who fills a $2\times 2$ square with his symbols wins. Can the player who starts always win?
2009 Canada National Olympiad, 1
Given an $m\times n$ grid with unit squares coloured either black or white, a black square in the grid is [i]stranded [/i]if there is some square to its left in the same row that is white and there is some square above it in the same column that is white.
Find a closed formula for the number of $2\times n$ grids with no stranded black square.
Note that $n$ is any natural number and the formula must be in terms of $n$ with no other variables.
2007 All-Russian Olympiad Regional Round, 9.3
$ 25$ boys and some girls came to the party and discovered an interesting property of their company. Take an arbitrary group of $ \geq 10$ boys and all the girls which are acquainted with at least one of them. Then in the joint group, the number of girls is by one greater than the number of boys. Prove that there exists a girl who is acquainted with at least $ 16$ boys.
2011 Olympic Revenge, 3
Let $E$ to be an infinite set of congruent ellipses in the plane, and $r$ a fixed line. It is known that each line parallel to $r$ intersects at least one ellipse belonging to $E$. Prove that there exist infinitely many triples of ellipses belonging to $E$, such that there exists a line that intersect the triple of ellipses.
1991 Canada National Olympiad, 5
The sides of an equilateral triangle $ABC$ are divided into $n$ equal parts $(n \geq 2) .$ For each point on a side, we draw the lines parallel to other sides of the triangle $ABC,$ e.g. for $n=3$ we have the following diagram:
[asy]
unitsize(150);
defaultpen(linewidth(0.7));
int n = 3; /* # of vertical lines, including AB */
pair A = (0,0), B = dir(-30), C = dir(30);
draw(A--B--C--cycle,linewidth(2)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0));
label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE);
for(int i = 1; i < n; ++i) {
draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n);
draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n);
draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n);
}
[/asy]
For each $n \geq 2,$ find the number of existing parallelograms.
2011 APMO, 4
Let $n$ be a fixed positive odd integer. Take $m+2$ [b]distinct[/b] points $P_0,P_1,\ldots ,P_{m+1}$ (where $m$ is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied:
1) $P_0=(0,1),P_{m+1}=(n+1,n)$, and for each integer $i,1\le i\le m$, both $x$- and $y$- coordinates of $P_i$ are integers lying in between $1$ and $n$ ($1$ and $n$ inclusive).
2) For each integer $i,0\le i\le m$, $P_iP_{i+1}$ is parallel to the $x$-axis if $i$ is even, and is parallel to the $y$-axis if $i$ is odd.
3) For each pair $i,j$ with $0\le i<j\le m$, line segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most $1$ point.
Determine the maximum possible value that $m$ can take.
2006 Baltic Way, 6
Determine the maximal size of a set of positive integers with the following properties:
$1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.
$2.$ No digit occurs more than once in the same integer.
$3.$ The digits in each integer are in increasing order.
$4.$ Any two integers have at least one digit in common (possibly at different positions).
$5.$ There is no digit which appears in all the integers.
2009 Middle European Mathematical Olympiad, 7
The numbers $ 0$, $ 1$, $ \dots$, $ n$ ($ n \ge 2$) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let $ g(n)$ be the smallest possible number of integers left on the blackboard at the end. Find $ g(n)$ for every $ n$.
2007 Indonesia TST, 4
Let $ X$ be a set of $ k$ vertexes on a plane such that no three of them are collinear. Let $ P$ be the family of all $ {k \choose 2}$ segments that connect each pair of points. Determine $ \tau(P)$.
1998 Romania Team Selection Test, 4
Consider in the plane a finite set of segments such that the sum of their lengths is less than $\sqrt{2}$. Prove that there exists an infinite unit square grid covering the plane such that the lines defining the grid do not intersect any of the segments.
[i]Vasile Pop[/i]
2006 Vietnam Team Selection Test, 3
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a [i]good[/i] number $m$.
2010 Contests, 1
Let $D$ be the set of all pairs $(i,j)$, $1\le i,j\le n$. Prove there exists a subset $S \subset D$, with $|S|\ge\left \lfloor\frac{3n(n+1)}{5}\right \rfloor$, such that for any $(x_1,y_1), (x_2,y_2) \in S$ we have $(x_1+x_2,y_1+y_2) \not \in S$.
(Peter Cameron)
2008 Stars Of Mathematics, 2
The $ 2^N$ vertices of the $ N$-dimensional hypercube $ \{0,1\}^N$ are labelled with integers from $ 0$ to $ 2^N \minus{} 1$, by, for $ x \equal{} (x_1,x_2,\ldots ,x_N)\in \{0,1\}^N$,
\[v(x) \equal{} \sum_{k \equal{} 1}^{N}x_k2^{k \minus{} 1}.\]
For which values $ n$, $ 2\leq n \leq 2^n$ can the vertices with labels in the set $ \{v|0\leq v \leq n \minus{} 1\}$ be connected through a Hamiltonian circuit, using edges of the hypercube only?
[i]E. Bazavan & C. Talau[/i]
2016 Croatia Team Selection Test, Problem 2
Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors.
Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.
1997 Hungary-Israel Binational, 1
Determine the number of distinct sequences of letters of length 1997 which use each of the letters $A$, $B$, $C$ (and no others) an odd number of times.