Found problems: 288
2019 China Girls Math Olympiad, 2
Find integers $a_1,a_2,\cdots,a_{18}$, s.t. $a_1=1,a_2=2,a_{18}=2019$, and for all $3\le k\le 18$, there exists $1\le i<j<k$ with $a_k=a_i+a_j$.
1996 All-Russian Olympiad Regional Round, 9.3
Let $a, b$ and $c$ be pairwise relatively prime natural numbers. Find all possible values of $\frac{(a + b)(b + c)(c + a)}{abc}$ if known what it is integer.
2000 Moldova Team Selection Test, 9
The sequence $x_{n}$ is de fined by:
$x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$
Prove that all members of the sequence are perfect squares.
2008 Mathcenter Contest, 4
Let $a,b$ and $c$ be positive integers that $$\frac{a\sqrt{3}+b}{b\sqrt3+c}$$ is a rational number, show that $$\frac{a^2+b^2+c^2}{a+b+ c}$$ is an integer.
[i](Anonymous314)[/i]
2013 Balkan MO Shortlist, N8
Suppose that $a$ and $b$ are integers. Prove that there are integers $c$ and $d$ such that $a+b+c+d=0$ and $ac+bd=0$, if and only if $a-b$ divides $2ab$.
2000 Nordic, 1
In how many ways can the number $2000$ be written as a sum of three positive, not necessarily different integers? (Sums like $1 + 2 + 3$ and $3 + 1 + 2$ etc. are the same.)
2002 Belarusian National Olympiad, 2
Given rational numbers $a_1,...,a_n$ such that $\sum_{i=1}^n \{ka_i\}<\frac{n}{2}$ for any positive integer $k$.
a) Prove that at least one of $a_1,...,a_n$ is integer.
b) Is the previous statement true, if the number $\frac{n}{2}$ is replaced by the greater number? (Here $\{x\}$ means a fractional part of $x$.)
(N. Selinger)
2018 German National Olympiad, 3
Given a positive integer $n$, Susann fills a square of $n \times n$ boxes. In each box she inscribes an integer, taking care that each row and each column contains distinct numbers. After this an imp appears and destroys some of the boxes.
Show that Susann can choose some of the remaining boxes and colour them red, satisfying the following two conditions:
1) There are no two red boxes in the same column or in the same row.
2) For each box which is neither destroyed nor coloured, there is a red box with a larger number in the same row or a red box with a smaller number in the same column.
[i]Proposed by Christian Reiher[/i]
2005 Mexico National Olympiad, 2
Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$.
(i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices.
(ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.
1996 Nordic, 2
Determine all real numbers $x$, such that $x^n+x^{-n}$ is an integer for all integers $n$.
2009 Bosnia And Herzegovina - Regional Olympiad, 4
Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers $\frac{x^2-1}{y+1}$ and $\frac{y^2-1}{x+1}$ are integers
2001 Nordic, 1
Let ${A}$ be a finite collection of squares in the coordinate plane such that the vertices of all squares that belong to ${A}$ are ${(m, n), (m + 1, n), (m, n + 1)}$, and ${(m + 1, n + 1)}$ for some integers ${m}$ and ${n}$. Show that there exists a subcollection ${B}$ of ${A}$ such that ${B}$ contains at least ${25 \% }$ of the squares in ${A}$, but no two of the squares in ${B}$ have a common vertex.
2015 Germany Team Selection Test, 2
A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$.
Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?
2015 Hanoi Open Mathematics Competitions, 12
Give a triangle $ABC$ with heights $h_a = 3$ cm, $h_b = 7$ cm and $h_c = d$ cm, where $d$ is an integer. Determine $d$.
2017 Singapore Senior Math Olympiad, 5
Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.
1957 Poland - Second Round, 1
Prove that if $ n $ is an integer, then
$$
\frac{n^5}{120} - \frac{n^3}{24} + \frac{n}{30}$$
is also an integer.
1994 Mexico National Olympiad, 2
The $12$ numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is $21$ or more.
2021 Indonesia TST, N
For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.
2005 Bosnia and Herzegovina Team Selection Test, 6
Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.
1934 Eotvos Mathematical Competition, 1
Let $n$ be a given positive integer and
$$A =\frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n- 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}$$
Prove that at least one term of the sequence $A, 2A,4A,8A,...,2^kA, ... $ is an integer.
2002 Korea Junior Math Olympiad, 5
Find all integer solutions to the equation
$$x^3+2y^3+4z^3+8xyz=0$$
1995 Nordic, 4
Show that there exist infinitely many mutually non- congruent triangles $T$, satisfying
(i) The side lengths of $T $ are consecutive integers.
(ii) The area of $T$ is an integer.
2020 Dutch IMO TST, 4
Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.
1997 Tournament Of Towns, (562) 3
All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is:
(a) an integer;
(b) the square of an integer.
(A Kanel)
2008 Abels Math Contest (Norwegian MO) Final, 1
Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$.
(a) Show that $s(n)$ is an integer whenever $n$ is an integer.
(b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?