Olympiad problems

2020 Brazil Team Selection Test, 3

Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n > m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.

1988 Poland - Second Round, 4

Tags: number theory , divide , divisible

Prove that for every natural number $ n $, the number $ n^{2n} - n^{n+2} + n^n - 1 $ is divisible by $ (n - 1 )^3 $.

2024 ELMO Shortlist, N7

Tags: number theory , polynomial

For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.) [i]Aprameya Tripathy[/i]

2018 Purple Comet Problems, 5

Tags: number theory

The positive integer $m$ is a multiple of $101$, and the positive integer $n$ is a multiple of $63$. Their sum is $2018$. Find $m - n$.

2025 Kyiv City MO Round 1, Problem 2

Tags: algebra , number theory , square

Prove that the number \[ 3 \underbrace{99\ldots9}_{2025} \underbrace{60\ldots01}_{2025} \] is a square of a positive integer.

1997 Belarusian National Olympiad, 1

Tags: number theory , sum of digits

$$Problem1:$$ A two-digit number which is not a multiple of $10$ is given. Assuming it is divisible by the sum of its digits, prove that it is also divisible by $3$. Does the statement hold for three-digit numbers as well?

2009 China Western Mathematical Olympiad, 1

Tags: modular arithmetic , number theory

Define a sequence $(x_{n})_{n\geq 1}$ by taking $x_{1}\in\left\{5,7\right\}$; when $k\ge 1$, $x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}$. Determine all possible last two digits of $x_{2009}$.

2008 District Olympiad, 3

Tags: floor function , number theory

Prove that if $ n\geq 4$, $ n\in\mathbb Z$ and $ \left \lfloor \frac {2^n}{n} \right\rfloor$ is a power of 2, then $ n$ is also a power of 2.

The Golden Digits 2024, P3

Tags: algebra , polynomial , number theory

Prove that there exist infinitely many positive integers $d$ such that we can find a polynomial $P\in\mathbb{Z}[x]$ of degree $d$ and $N\in\mathbb{N}$ such that for all integers $x>N$ and any prime $p$, we have $$\nu_p(P(x)^3+3P(x)^2-3)<\frac{d\cdot\log(x)}{2024^{2024}}.$$ [i]Proposed by Marius Cerlat[/i]

2018 India PRMO, 1

Tags: prime , divisor , number theory

A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?

2003 Spain Mathematical Olympiad, Problem 6

Tags: number theory

We string $2n$ white balls and $2n$ black balls, forming a continuous chain. Demonstrate that, in whatever order the balls are placed, it is always possible to cut a segment of the chain to contain exactly $n$ white balls and $n$ black balls.

2024 Azerbaijan BMO TST, 1

Tags: number theory

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

2023 India Regional Mathematical Olympiad, 2

Tags: number theory , prime number

Given a prime number $p$ such that $2p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36.

KoMaL A Problems 2018/2019, A. 751

Tags: number theory

Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c, 2^c, \cdots , n^c$ are integers. Prove that $c$ is an integer.

2003 All-Russian Olympiad Regional Round, 8.1

Tags: number theory

The numbers from $1$ to $10$ were divided into two groups so that the product of the numbers in the first group is completely divisible by the product of the numbers in the second. Which the smallest value can be for the quotient of the first product money for the second?

1979 IMO Longlists, 52

Tags: number theory

Let a real number $\lambda > 1$ be given and a sequence $(n_k)$ of positive integers such that $\frac{n_{k+1}}{n_k}> \lambda$ for $k = 1, 2,\ldots$ Prove that there exists a positive integer $c$ such that no positive integer $n$ can be represented in more than $c$ ways in the form $n = n_k + n_j$ or $n = n_r - n_s$.

2008 Bosnia Herzegovina Team Selection Test, 2

Tags: number theory

Find all pairs of positive integers $ m$ and $ n$ that satisfy (both) following conditions: (i) $ m^{2}\minus{}n$ divides $ m\plus{}n^{2}$ (ii) $ n^{2}\minus{}m$ divides $ n\plus{}m^{2}$

2021 Junior Balkan Team Selection Tests - Romania, P2

Tags: number theory , prime number

Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number.

2014 239 Open Mathematical Olympiad, 3

Tags: number theory

A natural number is called [i]good[/i] if it can be represented as sum of two coprime natural numbers, the first of which decomposes into odd number of primes (not necceserily distinct) and the second to even. Prove that there exist infinity many $n$ with $n^4$ being good.

2020 Jozsef Wildt International Math Competition, W16

Tags: number theory

Prove that: $$\left\lfloor10^{n+3}\cdot\sqrt{\overline{\underbrace{11\ldots1}_{2n\text{ times}}}}\right\rfloor=\overline{\underbrace{33\ldots3}_{2n\text{ times}}166}$$ [i]Proposed by Ovidiu Pop[/i]

2005 Tournament of Towns, 1

Tags: number theory

For which $n \ge 2$ can one find a sequence of distinct positive integers $a_1, a_2, \ldots , a_n$ so that the sum $$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots +\frac{a_n}{a_1}$$ is an integer? [i](3 points)[/i]

1995 Baltic Way, 2

Tags: number theory

Let $a$ and $k$ be positive integers such that $a^2+k$ divides $(a-1)a(a+1)$. Prove that $k\ge a$.

2005 India IMO Training Camp, 2

Tags: geometry , 3d geometry , number theory

Prove that one can find a $n_{0} \in \mathbb{N}$ such that $\forall m \geq n_{0}$, there exist three positive integers $a$, $b$ , $c$ such that (i) $m^3 < a < b < c < (m+1)^3$; (ii) $abc$ is the cube of an integer.

2016 May Olympiad, 1

Tags: perfect cube , number theory

Seven different positive integers are written on a sheet of paper. The result of the multiplication of the seven numbers is the cube of a whole number. If the largest of the numbers written on the sheet is $N$, determine the smallest possible value of $N$. Show an example for that value of $N$ and explain why $N$ cannot be smaller.

1975 IMO, 2

Tags: modular arithmetic , sequence , additive number theory , number theory

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$