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

1997 Slovenia National Olympiad, Problem 1
Tags: diophantine equation , number theory
Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.

2020-21 KVS IOQM India, 26
Tags: number theory
Let $a,b,c$ be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum $a + b + c$. Find this sum.

2015 AIME Problems, 3
Tags: number theory , prime number , factoring polynomials
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.

1998 Slovenia National Olympiad, Problem 4
Tags: number theory
Alf was attending an eight-year elementary school on Melmac. At the end of each school year, he showed the certificate to his father. If he was promoted, his father gave him the number of cats equal to Alf’s age times the number of the grade he passed. During elementary education, Alf failed one grade and had to repeat it. When he finished elementary education he found out that the total number of cats he had received was divisible by $1998$. Which grade did Alf fail?

2017 Hanoi Open Mathematics Competitions, 7
Tags: number theory , digit
Determine two last digits of number $Q = 2^{2017} + 2017^2$

1983 Bundeswettbewerb Mathematik, 3
Tags: number theory , digit
A real number is called [i]triplex[/i] if it has a decimal representation in which none of $0$ and $3$ different digit occurs. Prove that every positive real number is the sum of nine triplex numbers.

2008 Argentina Iberoamerican TST, 1
Tags: number theory
Find all integers $ x$ such that $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square It's a nice problem ...hope you enjoy it! Daniel

2017 Peru IMO TST, 12
Tags: number theory
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.

2001 Switzerland Team Selection Test, 10
Tags: subset , number theory , sum , power of 2
Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.

2016 Greece Team Selection Test, 1
Tags: number theory , sequence , divisibility
Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$. Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

2012 Dutch IMO TST, 3
Tags: diophantine equation , diophantine , number theory
Determine all pairs $(x, y)$ of positive integers satisfying $x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.

2021 Serbia National Math Olympiad, 1
Tags: number theory
Let $a>1$ and $c$ be natural numbers and let $b\neq 0$ be an integer. Prove that there exists a natural number $n$ such that the number $a^n+b$ has a divisor of the form $cx+1$, $x\in\mathbb{N}$.

2014 Cono Sur Olympiad, 4
Tags: number theory
Show that the number $n^{2} - 2^{2014}\times 2014n + 4^{2013} (2014^{2}-1)$ is not prime, where $n$ is a positive integer.

2010 Middle European Mathematical Olympiad, 12
Tags: number theory
We are given a positive integer $n$ which is not a power of two. Show that ther exists a positive integer $m$ with the following two properties: (a) $m$ is the product of two consecutive positive integers; (b) the decimal representation of $m$ consists of two identical blocks with $n$ digits. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 8)[/i]

2018 Purple Comet Problems, 10
Tags: number theory
Find the remainder when $11^{2018}$ is divided by $100$.

2021 South Africa National Olympiad, 3
Tags: perfect square , number theory , sum of squares , power of 2 , prime
Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.

2001 India IMO Training Camp, 2
Tags: modular arithmetic , number theory
Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that : \[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]

1979 Bundeswettbewerb Mathematik, 4
Tags: sequence , infinitely many solutions , number theory , last digit
An infinite sequence $p_1, p_2, p_3, \ldots$ of natural numbers in the decimal system has the following property: For every $i \in \mathbb{N}$ the last digit of $p_{i+1}$ is different from $9$, and by omitting this digit one obtains number $p_i$. Prove that this sequence contains infinitely many composite numbers.

2002 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , diophantine , diophantine equation , divisor
Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$. a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$. b) Find the number of all non-negative solutions of the above equation.

2018 CMI B.Sc. Entrance Exam, 3
Tags: functional equation , number theory , combinatorics
Let $f$ be a function on non-negative integers defined as follows $$f(2n)=f(f(n))~~~\text{and}~~~f(2n+1)=f(2n)+1$$ [b](a)[/b] If $f(0)=0$ , find $f(n)$ for every $n$. [b](b)[/b] Show that $f(0)$ cannot equal $1$. [b](c)[/b] For what non-negative integers $k$ (if any) can $f(0)$ equal $2^k$ ?

1996 Estonia National Olympiad, 1
Tags: prime , diophantine equation , diophantine , number theory
Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.

1992 IMO Longlists, 75
Tags: subset , number theory , sequence , floor function
A sequence $\{an\}$ of positive integers is defined by \[a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N\] Determine the positive integers that occur in the sequence.

2005 China Team Selection Test, 3
Tags: modular arithmetic , number theory , binomial expansion , fermat number , power of 2
$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.

2019 Polish Junior MO First Round, 3
Tags: inequalities , number theory
The integers $a, b, c$ are not $0$ such that $\frac{a}{b + c^2}=\frac{a + c^2}{b}$. Prove that $a + b + c \le 0$.

2018 Switzerland - Final Round, 3
Tags: number theory , least common multiple , lcm
Determine all natural integers $n$ for which there is no triplet $(a, b, c)$ of natural numbers such that: $$n = \frac{a \cdot \,\,lcm(b, c) + b \cdot lcm \,\,(c, a) + c \cdot lcm \,\, (a, b)}{lcm \,\,(a, b, c)}$$