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: 177

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 $a = b = 0$.

2016 Germany Team Selection Test, 2
Tags: inequalities , Integer , number theory , number theory unsolved , n-variable inequality
The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

2012 Dutch IMO TST, 3
Tags: number theory , Integer
Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.

2008 Postal Coaching, 1
Tags: Sequence , algebra , Integer , recurrence relation
Define a sequence $<x_n>$ by $x_0 = 0$ and $$\large x_n = \left\{ \begin{array}{ll} x_{n-1} + \frac{3^r-1}{2} & if \,\,n = 3^{r-1}(3k + 1)\\ & \\ x_{n-1} - \frac{3^r+1}{2} & if \,\, n = 3^{r-1}(3k + 2)\\ \end{array} \right. $$ where $k, r$ are integers. Prove that every integer occurs exactly once in the sequence.

2005 Bosnia and Herzegovina Junior BMO TST, 2
Tags: Perfect Square , Integer , number theory
Let n be a positive integer. Prove the following statement: ”If $2 + 2\sqrt{1 + 28n^2}$ is an integer, then it is the square of an integer.”

2018 Dutch BxMO TST, 2
Tags: geometry , Integer , Sides of a triangle , coprime
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.

2015 FYROM JBMO Team Selection Test, 1
Tags: number theory , Diophantine equation , Integer
Solve the equation $x^2+y^4+1=6^z$ in the set of integers.

1965 Polish MO Finals, 2
Tags: algebra , number theory , coprime , Integer
Prove that if the numbers $ x_1 $ and $ x_2 $ are roots of the equation $ x^2 + px - 1 = 0 $, where $ p $ is an odd number, then for every natural $n$number $ x_1^n + x_2^n $ and $ x_1^{n+1} + x_2^{n+1} $ are integer and coprime.

1986 All Soviet Union Mathematical Olympiad, 437
Tags: number theory , Integer , reciprocal , Sum
Prove that the sum of all numbers representable as $\frac{1}{mn}$, where $m,n$ -- natural numbers, $1 \le m < n \le1986$, is not an integer.

2014 Cuba MO, 4
Tags: number theory , Integer
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.

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)

1996 Nordic, 1
Tags: Integer , sum of digits , number theory
Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.

2016 Romania National Olympiad, 1
Tags: number theory , Integer
Find all non-negative integers $n$ so that $\sqrt{n + 3}+ \sqrt{n +\sqrt{n + 3}} $ is an integer.

2019 China Girls Math Olympiad, 2
Tags: Integer , algebra , number theory
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 $a_k=a_i+a_j$.

2021 Durer Math Competition Finals, 5
Tags: number theory , Integer
How many integers $1\le x \le 2021$ make the value of the expression $$\frac{2x^3 - 6x^2 - 3x -20}{5(x - 4)}$$ an integer?

1998 Singapore Senior Math Olympiad, 1
Tags: Integer , divisible , divides , number theory
Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.

2016 Germany Team Selection Test, 2
Tags: inequalities , Integer , number theory , number theory unsolved , n-variable inequality
The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

1918 Eotvos Mathematical Competition, 2
Tags: number theory , Integer , reciprocal
Find three distinct natural numbers such that the sum of their reciprocals is an integer.

2020 Tournament Of Towns, 3
Tags: Cyclic , inscribed , polygon , Integer , geometry , combinatorial geometry , combinatorics
Is it possible to inscribe an $N$-gon in a circle so that all the lengths of its sides are different and all its angles (in degrees) are integer, where a) $N = 19$, b) $N = 20$ ? Mikhail Malkin

2020 New Zealand MO, 1
Tags: number theory , Integer
What is the maximum integer $n$ such that $\frac{50!}{2^n}$ is an integer?

2002 Korea Junior Math Olympiad, 5
Tags: Integer , number theory
Find all integer solutions to the equation $$x^3+2y^3+4z^3+8xyz=0$$

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.

2024 Czech-Polish-Slovak Junior Match, 4
Tags: inequalities , inequalities proposed , Integer
Let $a,b,c$ be integers satisfying $a+b+c=1$ and $ab+bc+ca<abc$. Show that $ab+bc+ca<2abc$.

2016 Singapore Senior Math Olympiad, 4
Tags: Integer , geometry , integer area , integer sidelengths , perpendicular
Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90^o$ or $270^o$. If the lengths of its sides are odd integers, prove that its area is an even integer.

2020-IMOC, N1
Tags: IMOC , equation , Integer , number theory
$\textbf{N1.}$ Find all nonnegative integers $a,b,c$ such that \begin{align*} a^2+b^2+c^2-ab-bc-ca = a+b+c \end{align*} [i]Proposed by usjl[/i]