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: 106

2013 Nordic, 3
Tags: number theory , sequence , rational
Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$

2000 Regional Competition For Advanced Students, 4
Tags: number theory , algebra , rational , sequence , recurrence relation
We consider the sequence $\{u_n\}$ defined by recursion $u_{n+1} =\frac{u_n(u_n + 1)}{n}$ for $n \ge 1$. (a) Determine the terms of the sequence for $u_1 = 1$. (b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers. (c) Show that for every natural number $K$ there is a $u_1 > 1$ such that the first $K$ terms of the sequence are natural numbers.

1995 Yugoslav Team Selection Test, Problem 1
Tags: number theory , rational , rational number
Determine all triples $(x,y,z)$ of positive rational numbers with $x\le y\le z$ such that $x+y+z,\frac1x+\frac1y+\frac1z$, and xyz are natural numbers.

2017 Latvia Baltic Way TST, 13
Tags: rational , radical , algebra
Prove that the number $$\sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}}$$ is rational for all natural $n$.

2004 Estonia Team Selection Test, 4
Tags: algebra , trigonometry , sum , rational
Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$ For which positive integers $m$ is $f(m)$ rational?

1983 Kurschak Competition, 1
Tags: algebra , rational
Let $x, y$ and $z$ be rational numbers satisfying $$x^3 + 3y^3 + 9z^3 - 9xyz = 0.$$ Prove that $x = y = z = 0$.