This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 288

1997 Romania National Olympiad, 1
Tags: number theory , integer
Let $n_1 = \overline{abcabc}$ and $n_2= \overline{d00d}$ be numbers represented in the decimal system, with $a\ne 0$ and $d \ne 0$. a) Prove that $\sqrt{n_1}$ cannot be an integer. b) Find all positive integers $n_1$ and $n_2$ such that $\sqrt{n_1+n_2}$ is an integer number. c) From all the pairs $(n_1,n_2)$ such that $\sqrt{n_1 n_2}$ is an integer find those for which $\sqrt{n_1 n_2}$ has the greatest possible value

1996 All-Russian Olympiad Regional Round, 9.3
Tags: number theory , integer
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.

2011 Saudi Arabia Pre-TST, 2.1
Tags: number theory , integer
Let $n$ be a positive integer. Prove that the interval $I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right)$ does not contain any integer.

2002 Rioplatense Mathematical Olympiad, Level 3, 2
Tags: integer , positive integer , inequality , number theory , algebra
Let $\lambda$ be a real number such that the inequality $0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}$ holds for an infinite number of pairs $ (a, b)$ of positive integers. Prove that $\lambda \geq 5 $.

2015 Balkan MO Shortlist, N3
Tags: prime divisor , integer , number theory , divide , divisor
Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $ Let $s,t$ be two different positive integers with the following property: If $p$ is prime divisor of $s-t$, then $p$ divides $a-1$. Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer. (FYROM)

2021 Turkey Junior National Olympiad, 1
Tags: number theory , integer
Find all $(m, n)$ positive integer pairs such that both $\frac{3n^2}{m}$ and $\sqrt{n^2+m}$ are integers.

2017 Hanoi Open Mathematics Competitions, 10
Tags: trinomial , integer , root , algebra
Find all non-negative integers $a, b, c$ such that the roots of equations: $\begin{cases}x^2 - 2ax + b = 0 \\ x^2- 2bx + c = 0 \\ x^2 - 2cx + a = 0 \end{cases}$ are non-negative integers.

2015 Germany Team Selection Test, 2
Tags: integer , positive , consecutive , form , number theory
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?

2014 JBMO TST - Macedonia, 1
Tags: integer , inequality
Prove that $\frac{1}{1\times2013}+\frac{1}{2\times2012}+\frac{1}{3\times2011}+...+\frac{1}{2012\times2}+\frac{1}{2013\times1}<1$

2022 Mexican Girls' Contest, 6
Tags: number theory , composite number , integer
Let $a$ and $b$ be positive integers such that $$\frac{5a^4+a^2}{b^4+3b^2+4}$$ is an integer. Prove that $a$ is not a prime number.

2010 Dutch Mathematical Olympiad, 4
Tags: number theory , integer
(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.

1960 Kurschak Competition, 2
Tags: number theory , integer , sum
Let $a_1 = 1, a_2, a_3,...$: be a sequence of positive integers such that $$a_k < 1 + a_1 + a_2 +... + a_{k-1}$$ for all $k > 1$. Prove that every positive integer can be expressed as a sum of $a_i$s.

2009 Sharygin Geometry Olympiad, 7
Tags: geometry , vector , integer
Given points $O, A_1, A_2, ..., A_n$ on the plane. For any two of these points the square of distance between them is natural number. Prove that there exist two vectors $\vec{x}$ and $\vec{y}$, such that for any point $A_i$, $\vec{OA_i }= k\vec{x}+l \vec{y}$, where $k$ and $l$ are some integer numbers. (A.Glazyrin)

2003 Estonia National Olympiad, 2
Tags: number theory , integer
Find all possible integer values of $\frac{m^2+n^2}{mn}$ where m and n are integers.

2022 JBMO TST - Turkey, 1
Tags: number theory , divisibility , integer
For positive integers $a$ and $b$, if the expression $\frac{a^2+b^2}{(a-b)^2}$ is an integer, prove that the expression $\frac{a^3+b^3}{(a-b)^3}$ is an integer as well.

2002 Belarusian National Olympiad, 2
Tags: number theory , inequalities , integer
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)

1983 Brazil National Olympiad, 3
Tags: number theory , integer , sum
Show that $1 + 1/2 + 1/3 + ... + 1/n$ is not an integer for $n > 1$.

2009 Bosnia And Herzegovina - Regional Olympiad, 4
Tags: number theory , integer
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

2024 AMC 10, 7
Tags: integer
The product of three integers is $60$. What is the least possible positive sum of the three integers? $\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$

1992 Swedish Mathematical Competition, 1
Tags: number theory , integer
Is $\frac{19^{92} - 91^{29}}{90}$ an integer?

1985 Spain Mathematical Olympiad, 2
Tags: integer , subset , algebra
Determine if there exists a subset $E$ of $Z \times Z$ with the properties: (i) $E$ is closed under addition, (ii) $E$ contains $(0,0),$ (iii) For every $(a,b) \ne (0,0), E$ contains exactly one of $(a,b)$ and $-(a,b)$. Remark: We define $(a,b)+(a',b') = (a+a',b+b')$ and $-(a,b) = (-a,-b)$.

1957 Polish MO Finals, 3
Tags: number theory , integer
Prove that if the function $ ax^2 + bx + c $ takes an integer value for every integer value of the variable $ x $, then $ 2a $, $ a + b $, $ c $ are integers and vice versa.

1955 Moscow Mathematical Olympiad, 303
Tags: trinomial , integer , power , algebra , number theory
The quadratic expression $ax^2+bx+c$ is the $4$-th power (of an integer) for any integer $x$. Prove that $a = b = 0$.

2018 Rioplatense Mathematical Olympiad, Level 3, 5
Tags: positive integer , product , number theory , integer
Let $n$ be a positive integer. Find all $n$- rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$ \frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$ is integer for every integer $d\ge 0$

1957 Poland - Second Round, 1
Tags: number theory , integer
Prove that if $ n $ is an integer, then $$ \frac{n^5}{120} - \frac{n^3}{24} + \frac{n}{30}$$ is also an integer.