Olympiad problems

2017 All-Russian Olympiad, 5

$P(x)$ is polynomial with degree $n\geq 2$ and nonnegative coefficients. $a,b,c$ - sides for some triangle. Prove, that $\sqrt[n]{P(a)},\sqrt[n]{P(b)},\sqrt[n]{P(c)}$ are sides for some triangle too.

2006 Tournament of Towns, 6

Let $1 + 1/2 + 1/3 +... + 1/n = a_n/b_n$, where $a_n$ and $b_n$ are relatively prime. Show that there exist infinitely many positive integers $n$, such that $b_{n+1} < b_n$. (8)

sequence number theory decreasing sequences

Determine all possible (finite or infinite) values of \[\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),\] if $f:\mathbb{R} \to \mathbb{R}$ is a strictly decreasing continuous function satisfying \[f(f(x))^4-f(f(x))+f(x)=1\] for all $x \in \mathbb{R}$.

limit function functional equation decreasing bijection calculus functional equations limits

1999 Singapore Senior Math Olympiad, 3

Let $\{a_1,a_2,...,a_{100}\}$ be a sequence of $100$ distinct real numbers. Show that there exists either an increasing subsequence $a_{i_1}<a_{i_2}<...<a_{i_{10}}$ $(i_1<i_2<...<i_{10})$ of $10$ numbers, or a decreasing subsequence $ a_{j_1}>a_{j_2}>...>a_{j_{12}}$ $(j_1<j_2<...<j_{12})$ of $12$ numbers, or both.

subsequence inequalities increasing sequence decreasing algebra sequences

1999 Israel Grosman Mathematical Olympiad, 5

An infinite sequence of distinct real numbers is given. Prove that it contains a subsequence of $1999$ terms which is either monotonically increasing or monotonically decreasing.

subsequence increasing sequence decreasing algebra sequences

2017 All-Russian Olympiad, 5

2006 Tournament of Towns, 6

2018 VJIMC, 4

1999 Singapore Senior Math Olympiad, 3

1999 Israel Grosman Mathematical Olympiad, 5