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: 15460

1998 Swedish Mathematical Competition, 1
Tags: number theory , diophantine , diophantine equation
Find all positive integers $a, b, c$, such that $(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2$.

1997 APMO, 2
Tags: number theory
Find an integer $n$, where $100 \leq n \leq 1997$, such that \[ \frac{2^n+2}{n} \] is also an integer.

2007 China Team Selection Test, 1
Tags: modular arithmetic , number theory
Find all the pairs of positive integers $ (a,b)$ such that $ a^2 \plus{} b \minus{} 1$ is a power of prime number $ ; a^2 \plus{} b \plus{} 1$ can divide $ b^2 \minus{} a^3 \minus{} 1,$ but it can't divide $ (a \plus{} b \minus{} 1)^2.$

2024 Mexico National Olympiad, 2
Tags: number theory , divisor
Determine all pairs $(a, b)$ of integers that satisfy both: 1. $5 \leq b < a$ 2. There exists a natural number $n$ such that the numbers $\frac{a}{b}$ and $a-b$ are consecutive divisors of $n$, in that order. [b]Note:[/b] Two positive integers $x, y$ are consecutive divisors of $m$, in that order, if there is no divisor $d$ of $m$ such that $x < d < y$.

2018 China Team Selection Test, 6
Tags: pell equation , number theory , algebra
Find all pairs of positive integers $(x, y)$ such that $(xy+1)(xy+x+2)$ be a perfect square .

2017 SG Originals, N6
Tags: number theory , vieta jumping
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

2007 Singapore Junior Math Olympiad, 3
Tags: number theory , greatest common divisor
Let $n$ be a positive integer and $d$ be the greatest common divisor of $n^2+1$ and $(n + 1)^2 + 1$. Find all the possible values of $d$. Justify your answer.

2012 Danube Mathematical Competition, 3
Tags: number theory
Let $p$ and $q, p < q,$ be two primes such that $1 + p + p^2+...+p^m$ is a power of $q$ for some positive integer $m$, and $1 + q + q^2+...+q^n$ is a power of $p$ for some positive integer $n$. Show that $p = 2$ and $q = 2^t-1$ where $t$ is prime.

2016 SGMO, Q5
Tags: factorial , modular arithmetic , number theory
Let $d_{m} (n)$ denote the last non-zero digit of $n$ in base $m$ where $m,n$ are naturals. Given distinct odd primes $p_1,p_2,\ldots,p_k$, show that there exists infinitely many natural $n$ such that $$d_{2p_i} (n!) \equiv 1 \pmod {p_i}$$ for all $i = 1,2,\ldots,k$.

1991 Chile National Olympiad, 4
Tags: divide , divisible , number theory
Show that the expressions $2x + 3y$, $9x + 5y$ are both divisible by $17$, for the same values of $x$ and $y$.

2023 Auckland Mathematical Olympiad, 6
Tags: combinatorics , number theory
Suppose there is an infi nite sequence of lights numbered $1, 2, 3,...,$ and you know the following two rules about how the lights work: $\bullet$ If the light numbered $k$ is on, the lights numbered $2k$ and $2k + 1$ are also guaranteed to be on. $\bullet$ If the light numbered $k$ is off, then the lights numbered $4k + 1$ and $4k + 3$ are also guaranteed to be off. Suppose you notice that light number $2023$ is on. Identify all the lights that are guaranteed to be on?

2014 Kyiv Mathematical Festival, 4a
Tags: least common multiple , modular arithmetic , number theory
a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$ b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$

2020 German National Olympiad, 4
Tags: divisibility , number theory , divisor
Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.

2012 China Girls Math Olympiad, 8
Tags: number theory
Find the number of integers $k$ in the set $\{0, 1, 2, \dots, 2012\}$ such that $\binom{2012}{k}$ is a multiple of $2012$.

1998 IMO Shortlist, 4
Tags: number theory , integer sequence , calculate
A sequence of integers $ a_{1},a_{2},a_{3},\ldots$ is defined as follows: $ a_{1} \equal{} 1$ and for $ n\geq 1$, $ a_{n \plus{} 1}$ is the smallest integer greater than $ a_{n}$ such that $ a_{i} \plus{} a_{j}\neq 3a_{k}$ for any $ i,j$ and $ k$ in $ \{1,2,3,\ldots ,n \plus{} 1\}$, not necessarily distinct. Determine $ a_{1998}$.

MathLinks Contest 5th, 7.1
Tags: number theory
Prove that the numbers $${{2^n-1} \choose {i}}, i = 0, 1, . . ., 2^{n-1} - 1,$$ have pairwise different residues modulo $2^n$

2024 India Regional Mathematical Olympiad, 4
Tags: number theory
Let $n>1$ be a positive integer. Call a rearrangement $a_1,a_2, \cdots , a_n$ of $1,2, \cdots , n$ [i]nice[/i] if for every $k = 2 ,3, \cdots , n$, we have that $a_1^2 + a_2^2 + \cdots + a_k^2$ is [b]not[/b] divisible by $k$. Determine which positive integers $n>1$ have a [i]nice[/i] arrangement.

2018 China Team Selection Test, 2
Tags: number theory , arithmetic series , number of divisors
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.

2014 Swedish Mathematical Competition, 6
Tags: number theory , diophantine , diophantine equation
Determine all odd primes $p$ and $q$ such that the equation $x^p + y^q = pq$ at least one solution $(x, y)$ where $x$ and $y$ are positive integers.

2013 Iran Team Selection Test, 15
Tags: number theory
a) Does there exist a sequence $a_1<a_2<\dots$ of positive integers, such that there is a positive integer $N$ that $\forall m>N$, $a_m$ has exactly $d(m)-1$ divisors among $a_i$s? b) Does there exist a sequence $a_1<a_2<\dots$ of positive integers, such that there is a positive integer $N$ that $\forall m>N$, $a_m$ has exactly $d(m)+1$ divisors among $a_i$s?

2011 Indonesia TST, 4
Tags: number theory , recurrence relation , number theory with sequences , sequence
Let $a, b$, and $c$ be positive integers such that $gcd(a, b) = 1$. Sequence $\{u_k\}$, is given such that $u_0 = 0$, $u_1 = 1$, and u$_{k+2} = au_{k+1} + bu_k$ for all $k \ge 0$. Let $m$ be the least positive integer such that $c | u_m$ and $n$ be an arbitrary positive integer such that $c | u_n$. Show that $m | n$. [hide=PS.] There was a typo in the last line, as it didn't define what n does. Wording comes from [b]tst-2011-1.pdf[/b] from [url=https://sites.google.com/site/imoidn/idntst/2011tst]here[/url]. Correction was made according to #2[/hide]

2012 ITAMO, 4
Tags: modular arithmetic , quadratic , number theory
Let $x_1,x_2,x_3, \cdots$ be a sequence defined by the following recurrence relation: \[ \begin{cases}x_{1}&= 4\\ x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases} \] The first few terms of the sequence are $x_1=4,x_2=9,x_3=41 \cdots$ Find all pairs of positive integers $\{a,b\}$ such that $x_a x_b$ is a perfect square.

2001 Romania National Olympiad, 1
Tags: number theory
Show that there exist no integers $a$ and $b$ such that $a^3+a^2b+ab^2+b^3=2001$.

2019 Dutch IMO TST, 2
Tags: product , number theory
Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.

MOAA Team Rounds, 2022.5
Tags: number theory
Find the smallest positive integer that is equal to the sum of the product of its digits and the sum of its digits.