Found problems: 288
1976 Vietnam National Olympiad, 4
Find all three digit integers $\overline{abc} = n$, such that $\frac{2n}{3} = a! b! c!$
1998 Romania National Olympiad, 1
Let $n \ge 2$ be an integer and $M= \{1,2,\ldots,n\}.$ For each $k \in \{1,2,\ldots,n-1\}$ we define $$x_k= \frac{1}{n+1} \sum_{\substack{A \subset M \\ |A|=k}} (\min A + \max A).$$
Prove that the numbers $x_k$ are integers and not all of them are divisible by $4.$
[hide=Notations]$|A|$ is the cardinal of $A$
$\min A$ is the smallest element in $A$
$\max A$ is the largest element in $A$[/hide]
2006 Estonia Team Selection Test, 1
Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$.
a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$.
b) Find the sum of the other components of all such pairs of numbers.
2006 JBMO ShortLists, 15
Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.
2013 Bosnia And Herzegovina - Regional Olympiad, 2
If $x$ and $y$ are real numbers, prove that $\frac{4x^2+1}{y^2+2}$ is not integer
2020 Malaysia IMONST 1, 10
Given positive integers $a, b,$ and $c$ with $a + b + c = 20$.
Determine the number of possible integer values for $\frac{a + b}{c}.$
2004 Tournament Of Towns, 4
Arithmetical progression $a_1, a_2, a_3, a_4,...$ contains $a_1^2 , a_2^2$ and $a_3^2$ at some positions. Prove that all terms of this progression are integers.
1952 Moscow Mathematical Olympiad, 213
Given a geometric progression whose denominator $q$ is an integer not equal to $0$ or $-1$, prove that the sum of two or more terms in this progression cannot equal any other term in it.
2002 Korea Junior Math Olympiad, 5
Find all integer solutions to the equation
$$x^3+2y^3+4z^3+8xyz=0$$
2014 India PRMO, 13
For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?
2019 Czech-Polish-Slovak Junior Match, 1
Rational numbers $a, b$ are such that $a+b$ and $a^2+b^2$ are integers. Prove that $a, b$ are integers.
2016 Hanoi Open Mathematics Competitions, 15
Find all polynomials of degree $3$ with integer coeffcients such that $f(2014) = 2015, f(2015) = 2016$ and $f(2013) - f(2016)$ is a prime number.
2013 Tournament of Towns, 4
Is it true that every integer is a sum of finite number of cubes of distinct integers?
1995 Czech And Slovak Olympiad IIIA, 2
Find the positive real numbers $x,y$ for which $\frac{x+y}{2},\sqrt{xy},\frac{2xy}{x+y},\sqrt{\frac{x^2 +y^2}{2}}$ are integers whose sum is $66$.
2018 Dutch BxMO TST, 2
Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.
2023 ISL, N8
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
Mathley 2014-15, 2
Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$.
Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers.
Dang Hung Thang, University of Natural Sciences, Hanoi National University.
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 $n$.
2006 Tournament of Towns, 4
Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)
1998 Bosnia and Herzegovina Team Selection Test, 5
Let $a$, $b$ and $c$ be integers such that $$bc+ad=1$$ $$ac+2bd=1$$ Prove that $a^2+c^2=2b^2+2d^2$
2024 Thailand TST, 3
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
2022 Switzerland Team Selection Test, 6
Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.
2014 Cuba MO, 4
Find all positive integers $a, b$ such that the numbers $\frac{a^2b + a}{a^2 + b}$ and $\frac{ab^2 + b}{b^2 - a}$ are integers.
2015 Bangladesh Mathematical Olympiad, 2
[b][u]BdMO National Higher Secondary Problem 3[/u][/b]
Let $N$ be the number if pairs of integers $(m,n)$ that satisfies the equation $m^2+n^2=m^3$
Is $N$ finite or infinite?If $N$ is finite,what is its value?
2010 Contests, 4
(a) Determine all pairs $(x, y)$ of (real) numbers with $0 < x < 1$ and $0 <y < 1$ for which $x + 3y$ and $3x + y$ are both integer. An example is $(x,y) =( \frac{8}{3}, \frac{7}{8}) $, because $ x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} = 3$ and $ 3x+y = \frac98 + \frac78 =\frac{16}{8} = 2$.
(b) Determine the integer $m > 2$ for which there are exactly $119$ pairs $(x,y)$ with $0 < x < 1$ and $0 < y < 1$ such that $x + my$ and $mx + y$ are integers.
Remark: if $u \ne v,$ the pairs $(u, v)$ and $(v, u)$ are different.