Found problems: 396
2008 IMC, 4
Let $ \mathbb{Z}[x]$ be the ring of polynomials with integer coefficients, and let $ f(x), g(x) \in\mathbb{Z}[x]$ be nonconstant polynomials such that $ g(x)$ divides $ f(x)$ in $ \mathbb{Z}[x]$. Prove that if the polynomial $ f(x)\minus{}2008$ has at least 81 distinct integer roots, then the degree of $ g(x)$ is greater than 5.
2012 China Team Selection Test, 3
Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have
\[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]
2005 MOP Homework, 1
Two rooks on a chessboard are said to be attacking each other if they are placed in the same row or column of the board.
(a) There are eight rooks on a chessboard, none of them attacks any other. Prove that there is an even number of rooks on black fields.
(b) How many ways can eight mutually non-attacking rooks be placed on the 9 £ 9 chessboard so that all eight rooks are on squares of the same color.
2013 Bogdan Stan, 4
Consider $ 16 $ pairwise distinct natural numbers smaller than $ 1597. $
[b]a)[/b] Prove that among these, there are three numbers having the property that the sum of any two of them is bigger than the third.
[b]b)[/b] If one of these numbers is $ 1597, $ is still true the fact from subpoint [b]a)[/b]?
[i]Teodor Radu[/i]
2013 Serbia National Math Olympiad, 4
Determine all natural numbers $n$ for which there is a partition of $\{1,2,...,3n\}$ in $n$ pairwise disjoint subsets of the form $\{a,b,c\}$, such that numbers $b-a$ and $c-b$ are different numbers from the set $\{n-1, n, n+1\}$.
2021 Thailand Mathematical Olympiad, 4
Kan Krao Park is a circular park that has $21$ entrances and a straight line walkway joining each pair of two entrances. No three walkways meet at a single point. Some walkways are paved with bricks, while others are paved with asphalt.
At each intersection of two walkways, excluding the entrances, is planted lotus if the two walkways are paved with the same material, and is planted waterlily if the two walkways are paved with different materials.
Each walkway is decorated with lights if and only if the same type of plant is placed at least $45$ different points along that walkway. Prove that there are at least $11$ walkways decorated with lights and paved with the same material.
2009 South africa National Olympiad, 3
Ten girls, numbered from 1 to 10, sit at a round table, in a random order. Each girl then receives a new number, namely the sum of her own number and those of her two neighbours. Prove that some girl receives a new number greater than 17.
2012 China National Olympiad, 3
Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers.
[i]Proposed by Huawei Zhu[/i]
2011 IMO Shortlist, 2
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.
[i]Proposed by Luxembourg[/i]
PEN O Problems, 9
Let $n$ be an integer, and let $X$ be a set of $n+2$ integers each of absolute value at most $n$. Show that there exist three distinct numbers $a, b, c \in X$ such that $c=a+b$.
2000 Hungary-Israel Binational, 1
Let $A$ and $B$ be two subsets of $S = \{1, 2, . . . , 2000\}$ with $|A| \cdot |B| \geq 3999$. For a set $X$ , let $X-X$ denotes the set $\{s-t | s, t \in X, s \not = t\}$. Prove that $(A-A) \cap (B-B)$ is nonempty.
2009 Polish MO Finals, 4
Let $ x_1,x_2,..,x_n$ be non-negative numbers whose sum is $ 1$ . Show that there exist numbers $ a_1,a_2,\ldots ,a_n$ chosen from amongst $ 0,1,2,3,4$ such that $ a_1,a_2,\ldots ,a_n$ are different from $ 2,2,\ldots ,2$ and $ 2\leq a_1x_1\plus{}a_2x_2\plus{}\ldots\plus{}a_nx_n\leq 2\plus{}\frac{2}{3^n\minus{}1}$.
2011 USA Team Selection Test, 9
Determine whether or not there exist two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 < x < 1$ satisfies the following inequality:
\[\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}.\]
1961 AMC 12/AHSME, 39
Any five points are taken inside or on a square with side length $1$. Let $a$ be the [i]smallest[/i] possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is:
${{ \textbf{(A)}\ \sqrt{3}/3 \qquad\textbf{(B)}\ \sqrt{2}/2 \qquad\textbf{(C)}\ 2\sqrt{2}/3 \qquad\textbf{(D)}\ 1 }\qquad\textbf{(E)}\ \sqrt{2} } $
2006 Estonia Math Open Senior Contests, 6
Kati cut two equal regular $ n\minus{}gons$ out of paper. To the vertices of both $ n\minus{}gons$, she wrote the numbers 1 to $ n$ in some order. Then she stabbed a needle through the centres of these $ n\minus{}gons$ so that they could be rotated with respect to each other. Kati noticed that there is a position where the numbers at each pair of aligned vertices are different. Prove that the $ n\minus{}gons$ can be rotated to a position where at least two pairs of aligned vertices contain equal numbers.
1990 IMO Shortlist, 20
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.
1974 IMO Longlists, 43
An $(n^2+n+1) \times (n^2+n+1)$ matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed $(n + 1)(n^2 + n + 1).$
2013 China Team Selection Test, 2
For the positive integer $n$, define $f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|$. Let $\{n_i\}$ be a strictly increasing sequence of positive integers. $C$ is a constant such that $f(n_i)<\dfrac C{n_i^2}$ for all $i\in\{1,2,\ldots\}$. Show that there exists a real number $q>1$ such that $n_i\geqslant q^{i-1}$ for all $i\in\{1,2,\ldots \}$.
1996 Bosnia and Herzegovina Team Selection Test, 5
Group of $10$ people are buying books. We know the following:
$i)$ Every person bought four different books
$ii)$ Every two persons bought at least one book common for both of them
Taking in consideration book which was bought by maximum number of people, determine minimal value of that number
1999 IMC, 6
Let $A$ be a subset of $\mathbb{Z}/n\mathbb{Z}$ with at most $\frac{\ln(n)}{100}$ elements.
Define $f(r)=\sum_{s\in A} e^{\dfrac{2 \pi i r s}{n}}$. Show that for some $r \ne 0$ we have $|f(r)| \geq \frac{|A|}{2}$.
2014 Saudi Arabia BMO TST, 2
Prove that among any $16$ perfect cubes we can always find two cubes whose difference is divisible by $91$.
2003 Manhattan Mathematical Olympiad, 2
A tennis net is made of strings tied up together which make a grid consisting of small congruent squares as shown below.
[asy]
size(500);
xaxis(-50,50);
yaxis(-5,5);
add(shift(-50,-5)*grid(100,10));[/asy]
The size of the net is $100\times 10$ small squares. What is the maximal number of edges of small squares which can be cut without breaking the net into two pieces? (If an edge is cut, the cut is made in the middle, not at the ends.)
2010 Contests, 1
The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers:
\[a, b, c, a+b, b+c, c+a, a+b+c\]
belong to $S$.
2012 Korea National Olympiad, 4
Let $ p \equiv 3 \pmod{4}$ be a prime. Define $T = \{ (i,j) \mid i, j \in \{ 0, 1, \cdots , p-1 \} \} \smallsetminus \{ (0,0) \} $. For arbitrary subset $ S ( \ne \emptyset ) \subset T $, prove that there exist subset $ A \subset S $ satisfying following conditions:
(a) $ (x_i , y_i ) \in A ( 1 \le i \le 3) $ then $ p \not | x_1 + x_2 - y_3 $ or $ p \not | y_1 + y_2 + x_3 $.
(b) $ 8 n(A) > n(S) $
2024 Bangladesh Mathematical Olympiad, P4
Let $a_1, a_2, \ldots, a_{11}$ be integers. Prove that there exist numbers $b_1, b_2, \ldots, b_{11}$ such that
[list]
[*] $b_i$ is equal to $-1,0$ or $1$ for all $i \in \{1, 2,\dots, 11\}$.
[*] all numbers can't be zero at a time.
[*] the number $N=a_1b_1+a_2b_2+\ldots+a_{11}b_{11}$ is divisible by $2024$.
[/list]