Olympiad problems

Kvant 2022, M2714

Let $f{}$ and $g{}$ be polynomials with integers coefficients. The leading coefficient of $g{}$ is equal to 1. It is known that for infinitely many natural numbers $n{}$ the number $f(n)$ is divisible by $g(n)$ . Prove that $f(n)$ is divisible by $g(n)$ for all positive integers $n{}$ such that $g(n)\neq 0$. [i]From the folklore[/i]

Fractal Edition 1, P1

Tags: Divisibility , gyatt , Skibidi , no

Is the number $1234567890987654321$ prime?

1988 IMO Longlists, 10

Tags: algebra , polynomial , modular arithmetic , number theory , Divisibility , IMO Shortlist

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

2025 India National Olympiad, P1

Tags: INMO 2025 , Recurrence , Divisibility , Induct

Consider the sequence defined by $a_1 = 2$, $a_2 = 3$, and \[ a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1}, \] for all integers $k \geq 1$. Determine all positive integers $n$ such that \[ \frac{a_n}{n} \] is an integer. Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.

2008 Ukraine Team Selection Test, 10

Tags: modular arithmetic , number theory , Divisibility , IMO Shortlist , Hi

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

2016 Indonesia TST, 2

Tags: IMO Shortlist , number theory , Divisibility , Hi

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

1982 IMO, 1

Tags: number theory , equation , IMO , Diophantine equation , Divisibility , IMO 1982

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

2007 Alexandru Myller, 1

Tags: equations , Diophantine equation , number theory , Perfect Squares , Divisibility

Solve $ x^3-y^3=2xy+7 $ in integers.

2021 SAFEST Olympiad, 2

Tags: number theory , Sequence , Divisibility , IMO Shortlist , IMO Shortlist 2020

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

1992 IMO Shortlist, 19

Tags: algebra , polynomial , number theory , coefficients , Divisibility , IMO Shortlist

Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$

2004 German National Olympiad, 3

Tags: number theory , Divisibility , Digits , sum of digits

Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.

2021 Saudi Arabia IMO TST, 10

Tags: number theory , Sequence , Divisibility , IMO Shortlist , IMO Shortlist 2020

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

1990 IMO Longlists, 79

Tags: modular arithmetic , number theory , Divisibility , IMO 1990 , Orders And LTE , P3

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

2016 Kyiv Mathematical Festival, P5

Tags: Kyiv mathematical festival , combinatorics , Digits , Divisibility

On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?

2005 China Team Selection Test, 1

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2024 Kyiv City MO Round 1, Problem 1

Tags: number theory , algebra , Divisibility

Find all pairs of positive integers $(a, b)$ such that $4b - 1$ is divisible by $3a + 1$ and $3a - 1$ is divisible by $2b + 1$.

2019 China Team Selection Test, 4

Tags: number theory , Divisibility , China TST , Lte

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has $$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$ Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?

1998 Iran MO (3rd Round), 1

Tags: modular arithmetic , number theory , Divisibility , IMO Shortlist

Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

2013 IFYM, Sozopol, 1

Tags: algebra , Divisibility , Sequence

Let $u_1=1,u_2=2,u_3=24,$ and $u_{n+1}=\frac{6u_n^2 u_{n-2}-8u_nu_{n-1}^2}{u_{n-1}u_{n-2}}, n \geq 3.$ Prove that the elements of the sequence are natural numbers and that $n\mid u_n$ for all $n$.

2009 Serbia National Math Olympiad, 4

Tags: combinatorics , permutation , Divisibility , IMO Shortlist

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]