Olympiad problems

2011 Kazakhstan National Olympiad, 4

Tags: number theory

Prove that there are infinitely many natural numbers, the arithmetic mean and geometric mean of the divisors which are both integers.

2010 Chile National Olympiad, 1

Tags: number theory , perfect square

The integers $a, b$ satisfy the following identity $$2a^2 + a = 3b^2 + b.$$ Prove that $a- b$, $2a + 2b + 1$, and $3a + 3b + 1$ are perfect squares.

2013 ELMO Shortlist, 7

Tags: algebra , polynomial , modular arithmetic , number theory

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

2010 HMNT, 9

Tags: number theory

What is the sum of all numbers between $0$ and $511$ inclusive that have an even number of $1$s when written in binary?

2014 Junior Balkan Team Selection Tests - Romania, 3

Tags: sum , combinatorics , number theory , prime

Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.

2023 Paraguay Mathematical Olympiad, 4

Tags: number theory

We say that a positive integer is [i]Noble [/i] when: it is composite, it is not divisible by any prime number greater than $20$ and it is not divisible by any perfect cube greater than $1$. How many different Noble numbers are there?

1907 Eotvos Mathematical Competition, 1

Tags: number theory , rational roots , algebra

If $p$ and $q$ are odd integers, prove that the equation $$x^2 + 2px + 2q = 0$$ has no rational roots.

2011 Brazil Team Selection Test, 2

Tags: number theory , equation , multiplication

Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that \[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\] [i]Proposed by Daniel Brown, Canada[/i]

2002 Estonia National Olympiad, 3

Tags: number theory , digit

The teacher writes a $2002$-digit number consisting only of digits $9$ on the blackboard. The first student factors this number as $ab$ with $a > 1$ and $b > 1$ and replaces it on the blackboard by two numbers $a'$ and $b'$ with $|a-a'| = |b-b'| = 2$. The second student chooses one of the numbers on the blackboard, factors it as $cd$ with $c > 1$ and $d > 1$ and replaces the chosen number by two numbers $c'$ and $d'$ with $|c-c'| = |d-d'| = 2$, etc. Is it possible that after a certain number of students have been to the blackboard all numbers written there are equal to $9$?

2020 DMO Stage 1, 4.

Tags: number theory , sum of divisors

[b]Q[/b] Let $n\geq 2$ be a fixed positive integer and let $d_1,d_2,...,d_m$ be all positive divisors of $n$. Prove that: $$\frac{d_1+d_2+...+d_m}{m}\geq \sqrt{n+\frac{1}{4}}$$Also find the value of $n$ for which the equality holds. [i]Proposed by dangerousliri [/i]

2020 Federal Competition For Advanced Students, P2, 3

Tags: number theory , prime number , sequence

Let $a$ be a fixed positive integer and $(e_n)$ the sequence, which is defined by $e_0=1$ and $$ e_n=a + \prod_{k=0}^{n-1} e_k$$ for $n \geq 1$. Prove that (a) There exist infinitely many prime numbers that divide one element of the sequence. (b) There exists one prime number that does not divide an element of the sequence. (Theresia Eisenkölbl)

VI Soros Olympiad 1999 - 2000 (Russia), 8.3

Tags: number theory , digit

$72$ was added to the natural number $n$ and in the sum we got a number written in the same digits as the number $n$, but in the reverse order. Find all numbers $n$ that satisfy the given condition.

2011 Argentina National Olympiad, 5

Tags: number theory , divide , divisible

Find all integers $n$ such that $1<n<10^6$ and $n^3-1$ is divisible by $10^6 n-1$.

1997 Belarusian National Olympiad, 1

Tags: number theory

$$Problem 1$$ ;Find all composite numbers $n$ with the following property: For every proper divisor $d$ of $n$ (i.e. $1 < d < n$), it holds that $n-12 \geq d \geq n-20$.

2004 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , decimal representation , perfect square

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

2020 Greece Junior Math Olympiad, 3

Tags: number theory

Find all positive integers $x$, for which the equation $$a+b+c=xabc$$ has solution in positive integers. Solve the equation for these values of $x$

2008 Romania National Olympiad, 2

Tags: rectangle , ratio , number theory , greatest common divisor , relatively prime , geometry

A rectangle can be divided by parallel lines to its sides into 200 congruent squares, and also in 288 congruent squares. Prove that the rectangle can also be divided into 392 congruent squares.

2014 IMS, 11

Tags: number theory

Let the equation $a^2 + b^2 + 1=abc$ have answer in $\mathbb{N}$.Prove that $c=3$.

2012 Portugal MO, 3

Tags: number theory , greatest common divisor , floor function , ceiling function , combinatorics

Isabel wants to partition the set $\mathbb{N}$ of the positive integers into $n$ disjoint sets $A_{1}, A_{2}, \ldots, A_{n}$. Suppose that for each $i$ with $1\leq i\leq n$, given any positive integers $r, s\in A_{i}$ with $r\neq s$, we have $r+s\in A_{i}$. If $|A_{j}|=1$ for some $j$, find the greatest positive integer that may belong to $A_{j}$.

2013 Switzerland - Final Round, 9

Tags: number theory , diophantine , diophantine equation

Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$

2018 Peru Iberoamerican Team Selection Test, P3

Tags: number theory

For each positive integer $m$, be $P(m)$ the product of all the digits of $m$. It defines the succession $a_1,a_2, a_3\cdots, $ as follows: . $a_1$ is a positive integer less than 2018 . $a_{n+1}=a_n+P(a_n)$ for each integer $n\ge 1$ Prove that for every integer $n \ge1$ it is true that $a_n \le 2^{2018}.$

2003 USAMO, 1

Tags: induction , number theory

Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

2012 Thailand Mathematical Olympiad, 12

Tags: number theory , integer , perfect cube

Let $a, b, c$ be positive integers. Show that if $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer then $abc$ is a perfect cube.

2015 Princeton University Math Competition, A8

Tags: number theory

Let $n = 2^{2015} - 1$. For any integer $1 \le x < n$, let \[f_n(x) = \sum\limits_p s_p(n-x) + s_p(x) - s_p(n),\] where $s_q(k)$ denotes the sum of the digits of $k$ when written in base $q$ and the summation is over all primes $p$. Let $N$ be the number of values of $x$ such that $4 | f_n(x)$. What is the remainder when $N$ is divided by $1000?$

2004 Dutch Mathematical Olympiad, 5

Tags: number theory , right triangle

A right triangle with perpendicular sides $a$ and $b$ and hypotenuse $c$ has the following properties: $a = p^m$ and $b = q^n$ with $p$ and $q$ prime numbers and $m$ and $n$ positive integers, $c = 2k +1$ with $k$ a positive integer. Determine all possible values of $c$ and the associated values of $a$ and $b$.