Olympiad problems

2015 Saudi Arabia GMO TST, 4

Let $p$ be an odd prime number. Prove that there exists a unique integer $k$ such that $0 \le k \le p^2$ and $p^2$ divides $k(k + 1)(k + 2) ... (k + p - 3) - 1$. Malik Talbi

1975 Swedish Mathematical Competition, 5

Tags: number theory , divides

Show that $n$ divides $2^n + 1$ for infinitely many positive integers $n$.

1978 Chisinau City MO, 159

Tags: number theory , prime , divides , divisible , factorial

Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.

2015 Finnish National High School Mathematics Comp, 3

Tags: number theory , factorial , factor , exponential , divides

Determine the largest integer $k$ for which $12^k$ is a factor of $120! $

2000 Portugal MO, 3

Tags: number theory , divides , divisor

Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.

2014 Switzerland - Final Round, 5

Tags: number theory , divides , divisible

Let $a_1, a_2, ...$ a sequence of integers such that for every $n \in N$ we have: $$\sum_{d | n} a_d = 2^n.$$ Show for every $n \in N$ that $n$ divides $a_n$. Remark: For $n = 6$ the equation is $a_1 + a_2 + a_3 + a_6 = 2^6.$

1979 Chisinau City MO, 174

Tags: number theory , odd , divides , divisible

Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.

2022 Regional Olympiad of Mexico West, 5

Tags: floor function , number theory , divides

Determine all positive integers $n$ such that $\lfloor \sqrt{n} \rfloor - 1$ divides $n + 1$ and $\lfloor \sqrt{n} \rfloor +2$ divides $ n + 4$.

2005 Thailand Mathematical Olympiad, 11

Tags: number theory , divides

Find the smallest positive integer $x$ such that $2^{254}$ divides $x^{2005} + 1$.

1999 Abels Math Contest (Norwegian MO), 2b

Tags: divides , divisible , number theory

If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$

2019 Austrian Junior Regional Competition, 4

Tags: number theory , primes , divisible , divides

Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$ Prove that the sum of the four prime numbers is divisible by $60$. (Walther Janous)

1999 Denmark MO - Mohr Contest, 5

Tags: number theory , Digits , divides , divisible

Is there a number whose digits are only $1$'s and which is divided by $1999$?

1980 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , divides , divisible

Prove that for every nonnegative integer $ k$ there is a product $$(k + 1)(k + 2)...(k + 1980)$$ divisible by $ 1980^{197}$.

2013 Korea Junior Math Olympiad, 4

Tags: number theory , prime numbers , minimum , divides

Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.

2001 Estonia National Olympiad, 2

Tags: least common multiple , divides , divisor , number theory

A student wrote a correct addition operation $A/B+C/D = E/F$ on the blackboard, where both summands are irreducible and $F$ is the least common multiple of $B$ and $D$. After that, the student reduced the sum $E/F$ correctly by an integer $d$. Prove that $d$ is a common divisor of $B$ and $D$.

2022 Durer Math Competition Finals, 10

Tags: number theory , divides

The pair of positive integers $(a, b)$ is such that a does not divide $b$, $b$ does not divide a, both numbers are at most $100$, and they have the maximal possible number of common divisors. What is the largest possible value of $a \cdot· b$?

1965 Dutch Mathematical Olympiad, 2

Tags: number theory , Perfect Square , divides , divisible

Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$. Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.