Found problems: 1488
2003 Rioplatense Mathematical Olympiad, Level 3, 3
Without overlapping, hexagonal tiles are placed inside an isosceles right triangle of area $1$ whose hypotenuse is horizontal. The tiles are similar to the figure below, but are not necessarily all the same size.[asy]
unitsize(.85cm);
draw((0,0)--(1,0)--(1,1)--(2,2)--(-1,2)--(0,1)--(0,0),linewidth(1));
draw((0,2)--(0,1)--(1,1)--(1,2),dashed);
label("\footnotesize $a$",(0.5,0),S);
label("\footnotesize $a$",(0,0.5),W);
label("\footnotesize $a$",(1,0.5),E);
label("\footnotesize $a$",(0,1.5),E);
label("\footnotesize $a$",(1,1.5),W);
label("\footnotesize $a$",(-0.5,2),N);
label("\footnotesize $a$",(0.5,2),N);
label("\footnotesize $a$",(1.5,2),N);
[/asy] The longest side of each tile is parallel to the hypotenuse of the triangle, and the horizontal side of length $a$ of each tile lies between this longest side of the tile and the hypotenuse of the triangle. Furthermore, if the longest side of a tile is farther from the hypotenuse than the longest side of another tile, then the size of the first tile is larger or equal to the size of the second tile. Find the smallest value of $\lambda$ such that every such configuration of tiles has a total area less than $\lambda$.
1994 China Team Selection Test, 3
For any 2 convex polygons $S$ and $T$, if all the vertices of $S$ are vertices of $T$, call $S$ a sub-polygon of $T$.
[b]I. [/b]Prove that for an odd number $n \geq 5$, there exists $m$ sub-polygons of a convex $n$-gon such that they do not share any edges, and every edge and diagonal of the $n$-gon are edges of the $m$ sub-polygons.
[b]II.[/b] Find the smallest possible value of $m$.
2004 South East Mathematical Olympiad, 7
A tournament is held among $n$ teams, following such rules:
a) every team plays all others once at home and once away.(i.e. double round-robin schedule)
b) each team may participate in several away games in a week(from Sunday to Saturday).
c) there is no away game arrangement for a team, if it has a home game in the same week.
If the tournament finishes in 4 weeks, determine the maximum value of $n$.
2002 China Team Selection Test, 3
Given positive integer $ m \geq 17$, $ 2m$ contestants participate in a circular competition. In each round, we devide the $ 2m$ contestants into $ m$ groups, and the two contestants in one group play against each other. The groups are re-divided in the next round. The contestants compete for $ 2m\minus{}1$ rounds so that each contestant has played a game with all the $ 2m\minus{}1$ players. Find the least possible positive integer $ n$, so that there exists a valid competition and after $ n$ rounds, for any $ 4$ contestants, non of them has played with the others or there have been at least $ 2$ games played within those $ 4$.
2010 South East Mathematical Olympiad, 4
$A_1,A_2,\cdots,A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side.
1996 China National Olympiad, 1
$8$ singers take part in a festival. The organiser wants to plan $m$ concerts. For every concert there are $4$ singers who go on stage, with the restriction that the times of which every two singers go on stage in a concert are all equal. Find a schedule that minimises $m$.
2005 MOP Homework, 4
A convex $2004$-sided polygon $P$ is given such that no four vertices are cyclic. We call a triangle whose vertices are vertices of $P$ thick if all other $2001$ vertices of $P$ lie inside the circumcircle of the triangle, and thin if they all lie outside its circumcircle. Prove that the number of thick triangles is equal to the number of thin triangles.
2014 Iran MO (3rd Round), 5
An $n$-mino is a connected figure made by connecting $n$ $1 \times 1 $ squares. Two polyminos are the same if moving the first we can reach the second. For a polymino $P$ ,let $|P|$ be the number of $1 \times 1$ squares in it and $\partial P$ be number of squares out of $P$ such that each of the squares have at least on edge in common with a square from $P$.
(a) Prove that for every $x \in (0,1)$:\[\sum_P x^{|P|}(1-x)^{\partial P}=1\]
The sum is on all different polyminos.
(b) Prove that for every polymino $P$, $\partial P \leq 2|P|+2$
(c) Prove that the number of $n$-minos is less than $6.75^n$.
[i]Proposed by Kasra Alishahi[/i]
2004 Vietnam Team Selection Test, 1
Let us consider a set $S = \{ a_1 < a_2 < \ldots < a_{2004}\}$, satisfying the following properties: $f(a_i) < 2003$ and $f(a_i) = f(a_j) \quad \forall i, j$ from $\{1, 2,\ldots , 2004\}$, where $f(a_i)$ denotes number of elements which are relatively prime with $a_i$. Find the least positive integer $k$ for which in every $k$-subset of $S$, having the above mentioned properties there are two distinct elements with greatest common divisor greater than 1.
2005 MOP Homework, 4
Each of the players in a tennis tournament played one match against each of the others. If every player won at least one match, show that there are three players A,B, and C such that A beats B, B beats C, and C beats A. Such a triple of player is called a cycle. Determine the number of maximum cycles such a tournament can have.
1987 Austrian-Polish Competition, 4
Does the set $\{1,2,3,...,3000\}$ contain a subset $ A$ consisting of 2000 numbers that $x\in A$ implies $2x \notin A$ ?!! :?:
2006 Estonia National Olympiad, 1
We call a [i]ship[/i] a figure made up of unit squares connected by common edges.
Prove that if there is an odd number of possible different ships consisting of n unit
squares on a $ 10 \times 10$ board, then n is divisible by 4.
2013 Indonesia MO, 1
In a $4 \times 6$ grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.
2003 Korea - Final Round, 1
Some computers of a computer room have a following network. Each computers are connected by three cable to three computers. Two arbitrary computers can exchange data directly or indirectly (through other computers). Now let's remove $K$ computers so that there are two computers, which can not exchange data, or there is one computer left. Let $k$ be the minimum value of $K$. Let's remove $L$ cable from original network so that there are two computers, which can not exchange data. Let $l$ be the minimum value of $L$. Show that $k=l$.
2008 Tuymaada Olympiad, 3
100 unit squares of an infinite squared plane form a $ 10\times 10$ square. Unit segments forming these squares are coloured in several colours. It is known that the border of every square with sides on grid lines contains segments of at most two colours. (Such square is not necessarily contained in the original $ 10\times 10$ square.) What maximum number of colours may appear in this colouring?
[i]Author: S. Berlov[/i]
1994 All-Russian Olympiad, 4
Real numbers are written on the squares of an infinite grid. Two figures consisting of finitely many squares are given. They may be translated anywhere on the grid as long as their squares coincide with those of the grid. It is known that wherever the first figure is translated, the sum of numbers it covers is positive. Prove that the second figure can be translated so that the sum of the numbers it covers is also positive.
2005 Italy TST, 3
Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins.
$(a)$ Determine which player has a winning strategy for $N=2005$.
$(b)$ Determine which player has a winning strategy for $N=2004$.
$(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.
1989 China Team Selection Test, 4
$\forall n \in \mathbb{N}$, $P(n)$ denotes the number of the partition of $n$ as the sum of positive integers (disregarding the order of the parts), e.g. since $4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4$, so $P(4)=5$. "Dispersion" of a partition denotes the number of different parts in that partitation. And denote $q(n)$ is the sum of all the dispersions, e.g. $q(4)=1+2+2+1+1=7$. $n \geq 1$. Prove that
(1) $q(n) = 1 + \sum^{n-1}_{i=1} P(i).$
(2) $1 + \sum^{n-1}_{i=1} P(i) \leq \sqrt{2} \cdot n \cdot P(n)$.
2014 Iran MO (3rd Round), 4
Let $P$ be a regular $2n$-sided polygon. A [b]rhombus-ulation[/b] of $P$ is dividing $P$ into rhombuses such that no two intersect and no vertex of any rhombus is on the edge of other rhombuses or $P$.
(a) Prove that number of rhombuses is a function of $n$. Find the value of this function. Also find the number of vertices and edges of the rhombuses as a function of $n$.
(b) Prove or disprove that there always exists an edge $e$ of $P$ such that by erasing all the segments parallel to $e$ the remaining rhombuses are connected.
(c) Is it true that each two rhombus-ulations can turn into each other using the following algorithm multiple times?
Algorithm: Take a hexagon -not necessarily regular- consisting of 3 rhombuses and re-rhombus-ulate the hexagon.
(d) Let $f(n)$ be the number of ways to rhombus-ulate $P$. Prove that:\[\Pi_{k=1}^{n-1} ( \binom{k}{2} +1) \leq f(n) \leq \Pi_{k=1}^{n-1} k^{n-k} \]
2011 Postal Coaching, 4
In a lottery, a person must select six distinct numbers from $1, 2, 3,\dots, 36$ to put on a ticket. The lottery commitee will then draw six distinct numbers randomly from $1, 2, 3, \ldots, 36$. Any ticket with numbers not containing any of these $6$ numbers is a winning ticket. Show that there is a scheme of buying $9$ tickets guaranteeing at least one winning ticket, but $8$ tickets are not enough to guarantee a winning ticket in general.
2011 Tuymaada Olympiad, 1
Each real number greater than $1$ is coloured red or blue with both colours being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+b$ and $ab$ are of different colours.
2007 China Team Selection Test, 3
There are $ 63$ points arbitrarily on the circle $ \mathcal{C}$ with its diameter being $ 20$. Let $ S$ denote the number of triangles whose vertices are three of the $ 63$ points and the length of its sides is no less than $ 9$. Fine the maximum of $ S$.
2005 MOP Homework, 5
Determine if it is possible to choose nine points in the plane such that there are $n=10$ lines in the plane each of which passes through exactly three of the chosen points. What if $n=11$?
2014 ITAMO, 6
A $(2n + 1) \times (2n + 1)$ grid, with $n> 0$, is colored in such a way that each of the cell is white or black. A cell is called [i]special[/i] if there are at least $n$ other cells of the same color in its row, and at least another $n$ cells of the same color in its column.
(a) Prove that there are at least $2n + 1$ special boxes.
(b) Provide an example where there are at most $4n$ special cells.
(c) Determine, as a function of $n$, the minimum possible number of special cells.
2003 Tournament Of Towns, 4
Several squares on a $15 \times 15$ chessboard are marked so that a bishop placed on any square of the board attacks at least two of marked squares. Find the minimal number of marked squares.