Olympiad problems

2006 Korea Junior Math Olympiad, 5

Find all positive integers that can be written in the following way $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$ Also, $m,n$ are relatively prime positive integers.

1985 Spain Mathematical Olympiad, 4

Tags: number theory , positive integer , prime , sum of powers

Prove that for each positive integer $k $ there exists a triple $(a,b,c)$ of positive integers such that $abc = k(a+b+c)$. In all such cases prove that $a^3+b^3+c^3$ is not a prime.

2007 JBMO Shortlist, 4

Tags: algebra , positive integer

Let $a$ and $ b$ be positive integers bigger than $2$. Prove that there exists a positive integer $k$ and a sequence $n_1, n_2, ..., n_k$ consisting of positive integers, such that $n_1 = a,n_k = b$, and $(n_i + n_{i+1}) | n_in_{i+1}$ for all $i = 1,2,..., k - 1$

2012 Greece JBMO TST, 4

Tags: combinatorics , integer solutions , positive integer

Numbers $x,y,z$ are positive integers and satisfy the equation $x+y+z=2013$. (E) a) Find the number of the triplets $(x,y,z)$ that are solutions of the equation (E). b) Find the number of the solutions of the equation (E) for which $x=y$. c) Find the solution $(x,y,z)$ of the equation (E) for which the product $xyz$ becomes maximum.

2022 Pan-American Girls' Math Olympiad, 5

Tags: positive integer

Find all positive integers $k$ for which there exist $a$, $b$, and $c$ positive integers such that \[\lvert (a-b)^3+(b-c)^3+(c-a)^3\rvert=3\cdot2^k.\]

2016 Saudi Arabia Pre-TST, 2.3

Tags: positive integer , number theory

Let $u$ and $v$ be positive rational numbers with $u \ne v$. Assume that there are infinitely many positive integers $n$ with the property that $u^n - v^n$ are integers. Prove that $u$ and $v$ are integers.

2016 Peru IMO TST, 16

Tags: polynomial , divisibility , divisible , algebra , positive integer

Find all pairs $ (m, n)$ of positive integers that have the following property: For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.

2011 Bosnia and Herzegovina Junior BMO TST, 1

Tags: positive integer , number theory , equation

Solve equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{5}-\frac{1}{xy}$, where $x$ and $y$ are positive integers.

2008 India Regional Mathematical Olympiad, 5

Tags: number theory , divisibility , positive integer

Let $N$ be a ten digit positive integer divisible by $7$. Suppose the first and the last digit of $N$ are interchanged and the resulting number (not necessarily ten digit) is also divisible by $7$ then we say that $N$ is a good integer. How many ten digit good integers are there?

1997 Spain Mathematical Olympiad, 4

Tags: positive integer , number theory

Let $p$ be a prime number. Find all integers $k$ for which $\sqrt{k^2 -pk}$ is a positive integer.

2013 Dutch BxMO/EGMO TST, 2

Tags: positive integer , triple , combinatorics , number theory , sum

Consider a triple $(a, b, c)$ of pairwise distinct positive integers satisfying $a + b + c = 2013$. A step consists of replacing the triple $(x, y, z)$ by the triple $(y + z - x,z + x - y,x + y - z)$. Prove that, starting from the given triple $(a, b,c)$, after $10$ steps we obtain a triple containing at least one negative number.

2016 JBMO Shortlist, 1

Tags: positive integer , sum , combinatorics

Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}$ . Find the least positive integer $k$ making the number $k!\cdot S_{2016}$ an integer.

2015 Kyiv Math Festival, P3

Tags: positive integer , number theory , sum

Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

2008 Indonesia TST, 2

Tags: number theory , factorial , positive integer

Find all positive integers $1 \le n \le 2008$ so that there exist a prime number $p \ge n$ such that $$\frac{2008^p + (n -1)!}{n}$$ is a positive integer.

2003 Olympic Revenge, 7

Tags: positive integer , subset , max , combinatorics

Let $X$ be a subset of $R_{+}^{*}$ with $m$ elements. Find $X$ such that the number of subsets with the same sum is maximum.

2020 Greece National Olympiad, 4

Tags: composite number , number theory , composite , positive integer

Find all values of the positive integer $k$ that has the property: There are no positive integers $a,b$ such that the expression $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$ is a composite positive number.

2015 Dutch IMO TST, 5

Tags: positive integer , number theory , divide , min , max

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

2016 Irish Math Olympiad, 5

Tags: positive integer , inequalities , algebra

Let $a_1, a_2, ..., a_m$ be positive integers, none of which is equal to $10$, such that $a_1 + a_2 + ...+ a_m = 10m$. Prove that $(a_1a_2a_3 \cdot ...\cdot a_m)^{1/m} \le 3\sqrt{11}$.

1991 Nordic, 3

Tags: induction , inequalities , positive integer

Show that $ \frac{1}{2^2} +\frac{1}{3^2} +...+\frac{1}{n^2} <\frac{2}{3}$ for all $n \ge 2 $.

2015 Greece JBMO TST, 3

Tags: number theory , perfect square , positive integer

Prove that there is not a positive integer $n$ such that numbers $(n+1)2^n, (n+3)2^{n+2}$ are both perfect squares.

2021 Argentina National Olympiad, 2

Tags: perfect square , positive integer , number theory , prime number

Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$

2015 Dutch IMO TST, 5

Tags: positive integer , number theory , divide , min , max

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

2016 Romanian Master of Mathematics Shortlist, C1

Tags: positive integer , combinatorics , minimum

We start with any finite list of distinct positive integers. We may replace any pair $n, n + 1$ (not necessarily adjacent in the list) by the single integer $n-2$, now allowing negatives and repeats in the list. We may also replace any pair $n, n + 4$ by $n - 1$. We may repeat these operations as many times as we wish. Either determine the most negative integer which can appear in a list, or prove that there is no such minimum.

2016 Peru IMO TST, 4

Tags: positive integer , number theory , divide , min , max

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

1986 Brazil National Olympiad, 2

Tags: positive integer , number theory , combinatorics , consecutive

Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.