Found problems: 1800
2014 China Team Selection Test, 5
Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.
2013 Turkey Team Selection Test, 3
Some cities of a country consisting of $n$ cities are connected by round trip flights so that there are at least $k$ flights from any city and any city is reachable from any city. Prove that for any such flight organization these flights can be distributed among $n-k$ air companies so that one can reach any city from any city by using of at most one flight of each air company.
2010 Kyiv Mathematical Festival, 4
1) The numbers $1,2,3,\ldots,2010$ are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers $a$ and $b$ with $ab-a-b$ while the second player replaces them with $ab+a+b.$ The game ends when a single number remains on the blackboard. If this number is smaller than $1\cdot2\cdot3\cdot\ldots\cdot2010$ then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy?
2) The numbers $1,2,3,\ldots,2010$ are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers $a$ and $b$ with $ab-a-b+2$ while the second player replaces them with $ab+a+b.$ The game ends when a single number remains on the blackboard. If this number is smaller than $1\cdot2\cdot3\cdot\ldots\cdot2010$ then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy?
2010 Portugal MO, 3
On each day, more than half of the inhabitants of Évora eats [i]sericaia[/i] as dessert. Show that there is a group of 10 inhabitants of Évora such that, on each of the last 2010 days, at least one of the inhabitants ate [i]sericaia[/i] as dessert.
2008 All-Russian Olympiad, 8
We are given $ 3^{2k}$ apparently identical coins,one of which is fake,being lighter than the others. We also dispose of three apparently identical balances without weights, one of which is broken (and yields outcomes unrelated to the actual situations). How can we find the fake coin in $ 3k\plus{}1$ weighings?
2011 Switzerland - Final Round, 1
At a party, there are $2011$ people with a glass of fruit juice each sitting around a circular table. Once a second, they clink glasses obeying the following two rules:
(a) They do not clink glasses crosswise.
(b) At each point of time, everyone can clink glasses with at most one other person.
How many seconds pass at least until everyone clinked glasses with everybody else?
[i](Swiss Mathematical Olympiad 2011, Final round, problem 1)[/i]
2011 ELMO Shortlist, 7
Let $T$ be a tree. Prove that there is a constant $c>0$ (independent of $n$) such that every graph with $n$ vertices that does not contain a subgraph isomorphic to $T$ has at most $cn$ edges.
[i]David Yang.[/i]
2012 Iran MO (3rd Round), 2
Suppose $S$ is a convex figure in plane with area $10$. Consider a chord of length $3$ in $S$ and let $A$ and $B$ be two points on this chord which divide it into three equal parts. For a variable point $X$ in $S-\{A,B\}$, let $A'$ and $B'$ be the intersection points of rays $AX$ and $BX$ with the boundary of $S$. Let $S'$ be those points $X$ for which $AA'>\frac{1}{3} BB'$. Prove that the area of $S'$ is at least $6$.
[i]Proposed by Ali Khezeli[/i]
2002 Tuymaada Olympiad, 4
A rectangular table with 2001 rows and 2002 columns is partitioned into $1\times 2$ rectangles. It is known that any other partition of the table into $1\times 2$ rectangles contains a rectangle belonging to the original partition.
Prove that the original partition contains two successive columns covered by 2001 horizontal rectangles.
[i]Proposed by S. Volchenkov[/i]
1998 Brazil National Olympiad, 3
Two mathematicians, lost in Berlin, arrived on the corner of Barbarossa street with Martin Luther street and need to arrive on the corner of Meininger street with Martin Luther street. Unfortunately they don't know which direction to go along Martin Luther Street to reach Meininger Street nor how far it is, so they must go fowards and backwards along Martin Luther street until they arrive on the desired corner. What is the smallest value for a positive integer $k$ so that they can be sure that if there are $N$ blocks between Barbarossa street and Meininger street then they can arrive at their destination by walking no more than $kN$ blocks (no matter what $N$ turns out to be)?
2001 Romania National Olympiad, 3
Let $m,k$ be positive integers, $k<m$ and $M$ a set with $m$ elements. Prove that the maximal number of subsets $A_1,A_2,\ldots ,A_p$ of $M$ for which $A_i\cap A_j$ has at most $k$ elements, for every $1\le i<j\le p$, equals
\[ p_{max}=\binom{m}{0}+\binom{m}{1}+\binom{m}{2}+\ldots+\binom{m}{k+1}\]
2002 Iran Team Selection Test, 8
We call $A_{1},A_{2},A_{3}$ [i]mangool[/i] iff there is a permutation $\pi$ that $A_{\pi(2)}\not\subset A_{\pi(1)},A_{\pi(3)}\not\subset A_{\pi(1)}\cup A_{\pi(2)}$. A good family is a family of finite subsets of $\mathbb N$ like $X,A_{1},A_{2},\dots,A_{n}$. To each goo family we correspond a graph with vertices $\{A_{1},A_{2},\dots,A_{n}\}$. Connect $A_{i},A_{j}$ iff $X,A_{i},A_{j}$ are mangool sets. Find all graphs that we can find a good family corresponding to it.
2007 QEDMO 4th, 13
Let $n$ and $k$ be integers such that $0\leq k\leq n$. Prove that
$\sum_{u=0}^{k}\binom{n+u-1}{u}\binom{n}{k-2u}=\binom{n+k-1}{k}$.
Note that we use the following conventions:
$\binom{r}{0}=1$ for every integer $r$;
$\binom{u}{v}=0$ if $u$ is a nonnegative integer and $v$ is an integer satisfying $v<0$ or $v>u$.
Darij
2014 Contests, 3
Given are 100 different positive integers. We call a pair of numbers [i]good[/i] if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.)
[i]Proposed by Alexander S. Golovanov, Russia[/i]
2015 India Regional MathematicaI Olympiad, 2
Determine the number of $3-$digit numbers in base $10$ having at least one $5$ and at most one $3$.
2003 Romania Team Selection Test, 17
A permutation $\sigma: \{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ is called [i]straight[/i] if and only if for each integer $k$, $1\leq k\leq n-1$ the following inequality is fulfilled
\[ |\sigma(k)-\sigma(k+1)|\leq 2. \]
Find the smallest positive integer $n$ for which there exist at least 2003 straight permutations.
[i]Valentin Vornicu[/i]
1998 Brazil Team Selection Test, Problem 2
Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.
1998 Hong kong National Olympiad, 2
The underside of a pyramid is a convex nonagon , paint all the diagonals of the nonagon and all the ridges of the pyramid into white and black , prove : there exists a triangle ,the colour of its three sides are the same . ( PS:the sides of the nonagon is not painted. )
2004 Pre-Preparation Course Examination, 5
Let $ A\equal{}\{A_1,\dots,A_m\}$ be a family distinct subsets of $ \{1,2,\dots,n\}$ with at most $ \frac n2$ elements. Assume that $ A_i\not\subset A_j$ and $ A_i\cap A_j\neq\emptyset$ for each $ i,j$. Prove that:
\[ \sum_{i\equal{}1}^m\frac1{\binom{n\minus{}1}{|A_i|\minus{}1}}\leq1\]
2010 IMAR Test, 3
Given an integer $n\ge 2$, given $n+1$ distinct points $X_0,X_1,\ldots,X_n$ in the plane, and a positive real number $A$, show that the number of triangles $X_0X_iX_j$ of area $A$ does not exceed $4n\sqrt n$.
2010 International Zhautykov Olympiad, 2
In every vertex of a regular $n$ -gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $[\frac{n}{2}]$ positions clockwise from its initial position.
2024 Abelkonkurransen Finale, 3b
A $2024$-[i]table [/i]is a table with two rows and $2024$ columns containg all the numbers $1,2,\dots,4048$. Such a table is [i]evenly coloured[/i] if exactly half of the numbers in each row, and one number in each column, is coloured red. The [i]red sum[/i] in an evenly coloured $2024$-table is the sum of all the red numbers in the table.
Let $N$ be the largest number such that every $2024$-table has an even colouring with red sum $\ge N$. Determine $N$, and find the number of $2024$-tables such that every even colouring of the table has red sum $\le N$.
2010 Indonesia TST, 1
The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\minus{}b \ge 2$ and replace them with $ a\minus{}1$ and $ b\plus{}1$[/i]. Please, determine the maximum number of steps that can be done.
[i]Yudi Satria, Jakarta[/i]
2003 Baltic Way, 6
Let $n\ge 2$ and $d\ge 1$ be integers with $d\mid n$, and let $x_1,x_2,\ldots x_n$ be real numbers such that $x_1+x_2+\cdots + x_n=0$. Show that there are at least $\binom{n-1}{d-1}$ choices of $d$ indices $1\le i_1<i_2<\cdots <i_d\le n $ such that $x_{i_{1}}+x_{i_{2}}+\cdots +x_{i_{d}}\ge 0$.
2007 APMO, 5
A regular $ (5 \times 5)$-array of lights is defective, so that toggling the switch for one light causes each adjacent light in the same row and in the same column as well as the light itself to change state, from on to off, or from off to on. Initially all the lights are switched off. After a certain number of toggles, exactly one light is switched on. Find all the possible positions of this light.