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

2012 China Northern MO, 3
Tags: number theory , divide
Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that: (1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$ (2) If $m,n \in S$ , then $mn\in S$.

1986 IMO, 1
Tags: perfect square , quadratic , modular arithmetic , number theory , system of equations
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

VMEO III 2006, 12.3
Tags: number theory , floor function , algebra , function
Prove that for all $n\in\mathbb{Z}^+$, we have \[ \sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3} \]

1990 Irish Math Olympiad, 2
Tags: number theory
A sequence of primes $a_n$ is defined as follows: $a_1 = 2$, and, for all $n \geq 2$,$ a_n$ is the largest prime divisor of $a_1a_2...a_{n-1} + 1$. Prove that $a_n \neq 5$ for all n. I'm presuming it must involve proving it's never equal to 0 mod 5, but I don't know what to do. Thanks

2020 CHMMC Winter (2020-21), 2
Tags: number theory
Find the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$.

2003 Moldova Team Selection Test, 1
Tags: modular arithmetic , ratio , number theory , relatively prime
Each side of an arbitrarly triangle is divided into $ 2002$ congruent segments. After that, each vertex is joined with all "division" points on the opposite side. Prove that the number of the regions formed, in which the triangle is divided, is divisible by $ 6$. [i]Proposer[/i]: [b]Dorian Croitoru[/b]

2006 Bulgaria Team Selection Test, 2
Tags: number theory
[b] Problem 5. [/b]Denote with $d(a,b)$ the numbers of the divisors of natural $a$, which are greater or equal to $b$. Find all natural $n$, for which $d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$ [i]Ivan Landgev[/i]

1977 Bundeswettbewerb Mathematik, 3
Tags: integer , algebra , number theory
The number $50$ is written as a sum of several positive integers (not necessarily distinct) whose product is divisible by $100.$ What is the largest possible value of this product?

1984 IMO Longlists, 7
Tags: least common multiple , floor function , number theory
Prove that for any natural number $n$, the number $\dbinom{2n}{n}$ divides the least common multiple of the numbers $1, 2,\cdots, 2n -1, 2n$.

2015 IMO Shortlist, N5
Tags: number theory
Find all positive integers $(a,b,c)$ such that $$ab-c,\quad bc-a,\quad ca-b$$ are all powers of $2$. [i]Proposed by Serbia[/i]

2011 Junior Balkan Team Selection Tests - Moldova, 8
Tags: number theory
The natural numbers $m$ and $k$ satisfy the equality $$1001 \cdot 1002 \cdot ... \cdot 2010 \cdot 2011 = 2^m (2k + 1)$$. Find the number $m$.

2006 IMS, 2
Tags: induction , number theory
For each subset $C$ of $\mathbb N$, Suppose $C\oplus C=\{x+y|x,y\in C, x\neq y\}$. Prove that there exist a unique partition of $\mathbb N$ to sets $A$, $B$ that $A\oplus A$ and $B\oplus B$ do not have any prime numbers.

JOM 2014, 1.
Tags: number theory
Let $f(n)$ be the product of all factors of $n$. Find all natural numbers $n$ such that $f(n)$ is not a perfect power of $n$.

2008 Turkey Team Selection Test, 4
Tags: induction , number theory
The sequence $ (x_n)$ is defined as; $ x_1\equal{}a$, $ x_2\equal{}b$ and for all positive integer $ n$, $ x_{n\plus{}2}\equal{}2008x_{n\plus{}1}\minus{}x_n$. Prove that there are some positive integers $ a,b$ such that $ 1\plus{}2006x_{n\plus{}1}x_n$ is a perfect square for all positive integer $ n$.

PEN A Problems, 37
Tags: number theory , greatest common divisor , modular arithmetic , relatively prime , divisibility theory
If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.

2011 Akdeniz University MO, 1
Tags: number theory , prime number
Let $m,n$ positive integers and $p$ prime number with $p=3k+2$. If $p \mid {(m+n)^2-mn}$ , prove that $$p \mid m,n$$

2002 Singapore Team Selection Test, 3
Tags: divide , number theory
For every positive integer $n$, show that there is a positive integer $k$ such that $2k^2 + 2001k + 3 \equiv 0$ (mod $2^n$).

2012 India PRMO, 3
Tags: number theory
For how many pairs of positive integers $(x,y)$ is $x+3y=100$?

LMT Team Rounds 2021+, 2
Tags: number theory
How many integers of the form $n^{2023-n}$ are perfect squares, where $n$ is a positive integer between $1$ and $2023$ inclusive?

2001 Kazakhstan National Olympiad, 1
Tags: divide , divisible , number theory
Prove that there are infinitely many natural numbers $ n $ such that $ 2 ^ n + 3 ^ n $ is divisible by $ n $.

2011 Morocco TST, 1
Tags: modular arithmetic , number theory , equation , lte lemma
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2019 Moroccan TST, 4
Tags: number theory , prime number
Let $p$ be a prime number. Find all the positive integers $n$ such that $p+n$ divides $pn$

1998 Estonia National Olympiad, 1
Tags: number theory , greatest common divisor , divisor
Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.

2002 AMC 12/AHSME, 17
Tags: number theory , prime number
Several sets of prime numbers, such as $ \{ 7, 83, 421, 659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? $ \textbf{(A)}\ 193\qquad\textbf{(B)}\ 207\qquad\textbf{(C)}\ 225\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 447$

2018 Moldova EGMO TST, 4
Tags: number theory
Find all sets of positive integers $A=\big\{ a_1,a_2,...a_{19}\big\}$ which satisfy the following: $1\big) a_1+a_2+...+a_{19}=2017;$ $2\big) S(a_1)=S(a_2)=...=S(a_{19})$ where $S\big(n\big)$ denotes digit sum of number $n$.