This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 6

2008 Postal Coaching, 1
Tags: number theory , irrational , fraction , natural , algebra , floor function
For each positive $ x \in \mathbb{R}$, define $ E(x)=\{[nx]: n\in \mathbb{N}\}$ Find all irrational $ \alpha >1$ with the following property: If a positive real $ \beta$ satisfies $ E(\beta) \subset E(\alpha)$. then $ \frac{\beta}{\alpha}$ is a natural number.

2020 Abels Math Contest (Norwegian MO) Final, 2b
Tags: natural , parity , number theory
Assume that $a$ and $b$ are natural numbers with $a \ge b$ so that $ \sqrt{a+\sqrt{a^2-b^2}}$ is a natural number. Show that $a$ and $b$ have the same parity.

1995 Abels Math Contest (Norwegian MO), 3
Tags: sum , natural , algebra
Show that there exists a sequence $x_1,x_2,...$ of natural numbers in which every natural number occurs exactly once, such that the sums $\sum_{i=1}^n \frac{1}{x_i}$, $n = 1,2,3,...$, include all natural numbers.

2016 Hanoi Open Mathematics Competitions, 14
Tags: algebra , radical , natural
Given natural numbers $a,b$ such that $2015a^2+a = 2016b^2+b$. Prove that $\sqrt{a-b}$ is a natural number.

2014 Danube Mathematical Competition, 1
Tags: number theory , prime , sum , natural
Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers.

2004 Switzerland Team Selection Test, 12
Tags: number theory , natural
Find all natural numbers which can be written in the form $\frac{(a+b+c)^2}{abc}$ , where $a,b,c \in N$.