Olympiad problems

2020 Greece National Olympiad, 4

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.

2013 Dutch BxMO/EGMO TST, 2

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

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.

2008 India Regional Mathematical Olympiad, 3

Tags: inequality , positive integer , real number , inequalities

Prove that for every positive integer $n$ and a non-negative real number $a$, the following inequality holds: $$n(n+1)a+2n \geqslant 4\sqrt{a}(\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}).$$

2008 Indonesia TST, 2

Tags: positive integer , factorial , number theory

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.

2016 Lusophon Mathematical Olympiad, 1

Tags: positive integer , prime factor , product , number theory , coprime

Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?

2000 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: integer part , even , number theory , positive integer

Let $a, b$ and $c$ be positive integers such that $a^2 + b^2 + 1 = c^2$ . Prove that $[a/2] + [c / 2]$ is even. Note: $[x]$ is the integer part of $x$.

1990 Nordic, 4

Tags: positive integer , operation , number theory

It is possible to perform three operations $f, g$, and $h$ for positive integers: $f(n) = 10n, g(n) = 10n + 4$, and $h(2n) = n$; in other words, one may write $0$ or $4$ in the end of the number and one may divide an even number by $2$. Prove: every positive integer can be constructed starting from $4$ and performing a finite number of the operations $f, g,$ and $h$ in some order.

2006 Bosnia and Herzegovina Team Selection Test, 3

Tags: positive integer , fractional part , inequality , inequalities , number theory

Prove that for every positive integer $n$ holds inequality $\{n\sqrt{7}\}>\frac{3\sqrt{7}}{14n}$, where $\{x\}$ is fractional part of $x$.

2018 Danube Mathematical Competition, 3

Tags: positive integer , sum , product , rational , number theory

Find all the positive integers $n$ with the property: there exists an integer $k > 2$ and the positive rational numbers $a_1, a_2, ..., a_k$ such that $a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n$.

2015 Bosnia and Herzegovina Junior BMO TST, 4

Tags: positive integer , sum , remainder , combinatorics

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be positive integers from set $\{1, 2,..., n\}$ such that every number from this set occurs exactly once. Is it possible that numbers $a_1$, $a_1 + a_2 ,..., a_1 + a_2 + ... + a_n$ all have different remainders upon division by $n$, if: $a)$ $n=7$ $b)$ $n=8$

2016 JBMO Shortlist, 1

Tags: positive integer , combinatorics , sum

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.

2003 Bosnia and Herzegovina Team Selection Test, 3

Tags: algebra , inequality , positive integer

Prove that for every positive integer $n$ holds: $(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}$

2019 Poland - Second Round, 4

Tags: positive integer , divisibility , number theory

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 $a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}$.

2017 Czech-Polish-Slovak Match, 1

Tags: perfect square , positive integer , number theory

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)

1987 Spain Mathematical Olympiad, 2

Tags: positive integer , inequalities , binomial coefficient

Show that for each natural number $n > 1$ $1 \cdot \sqrt{{n \choose 1}}+ 2 \cdot \sqrt{{n \choose 2}}+...+n \cdot \sqrt{{n \choose n}} <\sqrt{2^{n-1}n^3}$

2002 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: positive integer , number theory , inequality , algebra , integer

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 $.

2016 Irish Math Olympiad, 5

Tags: positive integer , algebra , inequalities

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 $(a_1a_2a_3 \cdot ...\cdot a_m)^{1/m} \le 3\sqrt{11}$.

1982 Brazil National Olympiad, 2

Tags: positive integer , sequence , number theory

Any positive integer $n$ can be written in the form $n = 2^b(2c+1)$. We call $2c+1$ the[i] odd part[/i] of $n$. Given an odd integer $n > 0$, define the sequence $ a_0, a_1, a_2, ...$ as follows: $a_0 = 2^n-1, a_{k+1} $ is the [i]odd part[/i] of $3a_k+1$. Find $a_n$.

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.

2011 Bosnia and Herzegovina Junior BMO TST, 1

Tags: equation , positive integer , number theory

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

2010 Singapore Junior Math Olympiad, 3

Tags: positive integer , sum , combinatorics

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 $n$?

2015 JBMO Shortlist, C3

Tags: positive integer , array , table , combinatorics

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.

1987 Brazil National Olympiad, 3

Tags: positive integer , maximum , game strategy , minimum , combinatorics

Two players play alternately. The first player is given a pair of positive integers $(x_1, y_1)$. Each player must replace the pair $(x_n, y_n)$ that he is given by a pair of non-negative integers $(x_{n+1}, y_{n+1})$ such that $x_{n+1} = min(x_n, y_n)$ and $y_{n+1} = max(x_n, y_n)- k\cdot x_{n+1}$ for some positive integer $k$. The first player to pass on a pair with $y_{n+1} = 0$ wins. Find for which values of $x_1/y_1$ the first player has a winning strategy.

2015 Dutch IMO TST, 5

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

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

2006 MOP Homework, 7

Tags: number theory , positive integer , function

Let $n$ be a given integer greater than two, and let $S = \{1, 2,...,n\}$. Suppose the function $f : S^k \to S$ has the property that $f(a) \ne f(b)$ for every pair $a$ and $b$ of elements in $S^k$ with $a$ and $b$ differ in all components. Prove that $f$ is a function of one of its elements.