Olympiad problems

2013 Dutch BxMO/EGMO TST, 2

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.

2015 Dutch IMO TST, 5

Tags: number theory , divides , min , max , positive integers

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

2010 Singapore Junior Math Olympiad, 3

Tags: Sum , combinatorics , positive integers

Let $a_1, a_2, ..., a_n$ be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any $1 \le i < j \le n$, there exist $k,\ell$, both different from $i$ and $j$ such that $a_i + a_j = a_k + a_{\ell}$. What is the smallest possible value of $n$?

2022 Pan-American Girls' Math Olympiad, 5

Tags: PAGMO , positive integers

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 Peru IMO TST, 4

Tags: number theory , divides , min , max , positive integers

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

2017 Czech-Polish-Slovak Match, 1

Tags: Perfect Squares , number theory , positive integers

Find all positive real numbers $c$ such that there are infinitely many pairs of positive integers $(n,m)$ satisfying the following conditions: $n \ge m+c\sqrt{m - 1}+1$ and among numbers $n. n+1,.... 2n-m$ there is no square of an integer. (Slovakia)

2015 Dutch IMO TST, 5

Tags: number theory , divides , min , max , positive integers

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

2014 JBMO Shortlist, 6

Tags: number theory , consecutive , positive integers , squarefree , decimal representation

Vukasin, Dimitrije, Dusan, Stefan and Filip asked their teacher to guess three consecutive positive integers, after these true statements: Vukasin: " The sum of the digits of one number is prime number. The sum of the digits of another of the other two is, an even perfect number.($n$ is perfect if $\sigma\left(n\right)=2n$). The sum of the digits of the third number equals to the number of it's positive divisors". Dimitrije:"Everyone of those three numbers has at most two digits equal to $1$ in their decimal representation". Dusan:"If we add $11$ to exactly one of them, then we have a perfect square of an integer" Stefan:"Everyone of them has exactly one prime divisor less than $10$". Filip:"The three numbers are square free". Professor found the right answer. Which numbers did he mention?

2016 Romanian Master of Mathematics Shortlist, C1

Tags: combinatorics , minimum , positive integers

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.

2002 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: Inequality , Integers , positive integers , number theory , algebra

Let $\lambda$ be a real number such that the inequality $0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}$ holds for an infinite number of pairs $ (a, b)$ of positive integers. Prove that $\lambda \geq 5 $.

2015 Kyiv Math Festival, P3

Tags: number theory , positive integers , Sum , Kyiv mathematical festival

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 $1$?

2016 Irish Math Olympiad, 5

Tags: inequalities , positive integers , 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 $(a_1a_2a_3 \cdot ...\cdot a_m)^{1/m} \le 3\sqrt{11}$.

2013 Dutch BxMO/EGMO TST, 2

Tags: Triple , positive integers , 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.

2003 Olympic Revenge, 7

Tags: Subsets , max , combinatorics , positive integers

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

2021 Argentina National Olympiad, 2

Tags: number theory , prime numbers , Perfect Squares , positive integers , contests , Argentina

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 $1000\leq m \leq 2021.$

2018 Lusophon Mathematical Olympiad, 4

Tags: Diophantine equation , positive integers , number theory

Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.

1983 Austrian-Polish Competition, 4

Tags: number theory , positive integers , Sets , partition

The set $N$ has been partitioned into two sets A and $B$. Show that for every $n \in N$ there exist distinct integers $a, b > n$ such that $a, b, a + b$ either all belong to $A$ or all belong to $B$.

2013 JBMO Shortlist, 5

Tags: number theory , Diophantine equation , positive integers

Solve in positive integers: $\frac{1}{x^2}+\frac{y}{xz}+\frac{1}{z^2}=\frac{1}{2013}$ .

2006 Korea Junior Math Olympiad, 2

Tags: number theory , coprime , positive integers , Integers

Find all positive integers that can be written in the following way $\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}$ . Also, $a,b, c$ are positive integers that are pairwise relatively prime.

2015 Kyiv Math Festival, P3

Tags: number theory , positive integers , Sum , Kyiv mathematical festival

Is it true that every positive integer greater than $50$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

2018 Federal Competition For Advanced Students, P1, 4

Tags: number theory , positive integers

Let $M$ be a set containing positive integers with the following three properties: (1) $2018 \in M$. (2) If $m \in M$, then all positive divisors of m are also elements of $M$. (3) For all elements $k, m \in M$ with $1 < k < m$, the number $km + 1$ is also an element of $M$. Prove that $M = Z_{\ge 1}$. [i](Proposed by Walther Janous)[/i]

2016 JBMO Shortlist, 1

Tags: Sum , combinatorics , positive integers

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.

2019 Poland - Second Round, 4

Tags: number theory , positive integers , Divisibility

Let $a_1, a_2, \ldots, a_n$ ($n\ge 3$) be positive integers such that $gcd(a_1, a_2, \ldots, a_n)=1$ and for each $i\in \lbrace 1,2,\ldots, n \rbrace$ we have $a_i|a_1+a_2+\ldots+a_n$. Prove that $a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}$.

2015 JBMO Shortlist, C3

Tags: combinatorics , positive integers , Array , table

Positive integers are put into the following table. \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline 4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline 7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline 11 & 17 & 24 & 32 & 41 & & & & & \\ \hline 16 & 23 & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline \end{tabular} Find the number of the line and column where the number $2015$ stays.

2020 Greece National Olympiad, 4

Tags: composite number , positive integers , number theory , Composite

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.