Olympiad problems

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Found problems: 12

2015 Germany Team Selection Test, 2

Tags: form , integer , positive , consecutive , number theory

A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$. Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?

2013 Dutch Mathematical Olympiad, 4

Tags: equation , product , positive , number theory , divisor

For a positive integer n the number $P(n)$ is the product of the positive divisors of $n$. For example, $P(20) = 8000$, as the positive divisors of $20$ are $1, 2, 4, 5, 10$ and $20$, whose product is $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000$. (a) Find all positive integers $n$ satisfying $P(n) = 15n$. (b) Show that there exists no positive integer $n$ such that $P(n) = 15n^2$.

2015 Germany Team Selection Test, 2

Tags: integer , positive , form , consecutive , number theory

A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$. Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?

1999 North Macedonia National Olympiad, 1

Tags: combinatorics , sum , positive , algebra

In a set of $21$ real numbers, the sum of any $10$ numbers is less than the sum of the remaining $11$ numbers. Prove that all the numbers are positive.

2017 Czech-Polish-Slovak Match, 3

Tags: algebra , functional equation , positive , find all functions

Find all functions ${f : (0, +\infty) \rightarrow R}$ satisfying $f(x) - f(x+ y) = f \left( \frac{x}{y}\right) f(x + y)$ for all $x, y > 0$. (Austria)

1999 Nordic, 4

Tags: positive , inequalities

Let $a_1, a_2, . . . , a_n$ be positive real numbers and $n \ge 1$. Show that $n (\frac{1}{a_1}+...+\frac{1}{a_n}) \ge (\frac{1}{1+a_1}+...+\frac{1}{1+a_n})(n+\frac{1}{a_1}+...+\frac{1}{a_n})$ When does equality hold?

KoMaL A Problems 2017/2018, A. 713

Tags: sequence , positive , algebra

We say that a sequence $a_1,a_2,\cdots$ is [i]expansive[/i] if for all positive integers $j,\; i<j$ implies $|a_i-a_j|\ge \tfrac 1j$. Find all positive real numbers $C$ for which one can find an expansive sequence in the interval $[0,C]$.

2011 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , factor , positive

For every positive integer $n$ let $\tau (n)$ denote the number of its positive factors. Determine all $n \in N$ that satisfy the equality $\tau (n) = \frac{n}{3}$

1990 Czech and Slovak Olympiad III A, 2

Tags: triangle inequality , parameter , positive , quadratic , algebra , inequalities

Determine all values $\alpha\in\mathbb R$ with the following property: if positive numbers $(x,y,z)$ satisfy the inequality \[x^2+y^2+z^2\le\alpha(xy+yz+zx),\] then $x,y,z$ are sides of a triangle.

1992 Austrian-Polish Competition, 1

Tags: divisor , number theory , sum , positive , product , even

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

2009 Kyiv Mathematical Festival, 2

Tags: inequalities , algebra , three variable inequality , positive

Let $x,y,z$ be positive numebrs such that $x+y+z\le x^3+y^3+z^3$. Is it true that a) $x^2+y^2+z^2 \le x^3+y^3+z^3$ ? b) $x+y+z\le x^2+y^2+z^2$ ?

2015 Bundeswettbewerb Mathematik Germany, 2

Tags: number theory , sum , integer , positive

A sum of $335$ pairwise distinct positive integers equals $100000$. a) What is the least number of uneven integers in that sum? b) What is the greatest number of uneven integers in that sum?