Found problems: 1800
2014 Postal Coaching, 4
Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.
2006 India IMO Training Camp, 2
Let $p$ be a prime number and let $X$ be a finite set containing at least $p$ elements. A collection of pairwise mutually disjoint $p$-element subsets of $X$ is called a $p$-family. (In particular, the empty collection is a $p$-family.) Let $A$(respectively, $B$) denote the number of $p$-families having an even (respectively, odd) number of $p$-element subsets of $X$. Prove that $A$ and $B$ differ by a multiple of $p$.
2006 IMS, 1
Prove that for each $m\geq1$: \[\sum_{|k|<\sqrt m}\binom{2m}{m+k}\geq 2^{2m-1}\] [hide="Hint"]Maybe probabilistic method works[/hide]
2003 Iran MO (3rd Round), 12
There is a lamp in space.(Consider lamp a point)
Do there exist finite number of equal sphers in space that the light of the lamp can not go to the infinite?(If a ray crash in a sphere it stops)
2008 Macedonia National Olympiad, 4
We call an integer $ n > 1$ [i]good[/i] if, for any natural numbers $ 1 \le b_1, b_2, \ldots , b_{n\minus{}1} \le n \minus{} 1$ and any $ i \in \{0, 1, \ldots , n \minus{} 1\}$, there is a subset $ I$ of $ \{1, \ldots , n \minus{} 1\}$ such that $ \sum_{k\in I} b_k \equiv i \pmod n$. (The sum over the empty set is zero.) Find all good numbers.
2010 Turkey Junior National Olympiad, 3
In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.
1982 IMO Longlists, 20
Consider a cube $C$ and two planes $\sigma, \tau$, which divide Euclidean space into several regions. Prove that the interior of at least one of these regions meets at least three faces of the cube.
2009 Iran Team Selection Test, 12
$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that :
$ |T| \leq \frac {4}{9}n + \log_2 n + 2$
1986 IMO Longlists, 27
In an urn there are n balls numbered $1, 2, \cdots, n$. They are drawn at random one by one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum?
A term in a sequence is a local maximum if it is greater than all its neighbors.
2007 Junior Balkan Team Selection Tests - Romania, 3
A rectangularly paper is divided in polygons areas in the following way: at every step one of the existing surfaces is cut by a straight line, obtaining two new areas. Which is the minimum number of cuts needed such that between the obtained polygons there exists $251$ polygons with $11$ sides?
2006 India IMO Training Camp, 1
Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1,2,3,\cdots,n)$ which satisfy the condition $\sigma(j)+\sigma^{-1}(j)=n+1$ for all $j \in \{1,2,3,\cdots,n\}$.
2012 Pre - Vietnam Mathematical Olympiad, 3
In a country, there are some cities and the city named [i]Ben Song[/i] is capital. Each cities are connected with others by some two-way roads. One day, the King want to choose $n$ cities to add up with [i]Ben Song[/i] city to establish an [i]expanded capital[/i] such that the two following condition are satisfied:
(i) With every two cities in [i]expanded capital[/i], we can always find a road connecting them and this road just belongs to the cities of [i]expanded capital[/i].
(ii) There are exactly $k$ cities which do not belong to [i]expanded capital[/i] have the direct road to at least one city of [i]expanded capital[/i].
Prove that there are at most $\binom{n+k}{k}$ options to expand the capital for the King.
2014 District Olympiad, 2
We call a nonempty set $M$ good if its elements are positive integers, each
having exactly $4$ divisors. If the good set $M$ has $n$ elements, we denote by
$S_{M}$ the sum of all $4n$ divisors of its members (the sum may contain
repeating terms).
a) Prove that $A=\{2\cdot37,19\cdot37,29\cdot37\}$ is good and $S_{A}=2014$.
b) Prove that if the set $B$ is good and $8\in B$, then $S_{B}\neq2014$.
2011 ELMO Shortlist, 3
Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows.
(Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.)
[i]Linus Hamilton.[/i]
2007 Junior Balkan Team Selection Tests - Romania, 2
There are given the integers $1 \le m < n$. Consider the set $M = \{ (x,y);x,y \in \mathbb{Z_{+}}, 1 \le x,y \le n \}$. Determine the least value $v(m,n)$ with the property that for every subset $P \subseteq M$ with $|P| = v(m,n)$ there exist $m+1$ elements $A_{i}= (x_{i},y_{i}) \in P, i = 1,2,...,m+1$, for which the $x_{i}$ are all distinct, and $y_{i}$ are also all distinct.
2008 Cuba MO, 9
Today was realized the National Olimpiad in Cuba, this is the 3rd problem of the second day:
Prof that we can color the lattice points in the plane with two color so that every rectangle with vertices in the lattice points and edges parallels to the co-ordinate axis that have area 2^n is not monocromatic [/img]
2013 Iran MO (2nd Round), 2
Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero.
([i]Hint[/i]: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)
2010 Contests, 3
A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.
2006 Indonesia MO, 6
Every phone number in an area consists of eight digits and starts with digit $ 8$. Mr Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr Edy gets a phone number which consists of at most five different digits?
2012 Indonesia TST, 2
An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.
2004 CentroAmerican, 1
On a whiteboard, the numbers $1$ to $9$ are written. Players $A$ and $B$ take turns, and $A$ is first. Each player in turn chooses one of the numbers on the whiteboard and removes it, along with all multiples (if any). The player who removes the last number loses.
Determine whether any of the players has a winning strategy, and explain why.
2006 Kyiv Mathematical Festival, 2
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
The number $123456789$ is written on the blackboard. At each step it is allowed to choose its digits $a$ and $b$ of the same parity and to replace each of them by $\frac{a+b}{2}.$ Is it possible to obtain a number larger then
a)$800000000$; b)$880000000$ by such replacements?
2009 Indonesia TST, 2
Consider the following array:
\[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots
\] Find the 5-th number on the $ n$-th row with $ n>5$.
1996 Brazil National Olympiad, 3
Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$. For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$. Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$.
2002 Tournament Of Towns, 6
There's a large pile of cards. On each card a number from $1,2,\ldots n$ is written. It is known that sum of all numbers on all of the cards is equal to $k\cdot n!$ for some $k$. Prove that it is possible to arrange cards into $k$ stacks so that sum of numbers written on the cards in each stack is equal to $n!$.