Olympiad problems

1995 Poland - Second Round, 3

Let $a,b,c,d$ be positive irrational numbers with $a+b = 1$. Show that $c+d = 1$ if and only if $[na]+[nb] = [nc]+[nd]$ for all positive integers $n$.

1980 All Soviet Union Mathematical Olympiad, 303

Tags: limit , irrational , sequence , digit , algebra

The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$. a) Prove that the sequence $x_i$ has limit. b) Can this limit be irrational if we have started with the rational number? c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.

1977 Chisinau City MO, 153

Tags: algebra , irrational , trigonometry

Prove that the number $\tan \frac{\pi}{3^n}$ is irrational for any natural $n$.

2019 Dürer Math Competition (First Round), P4

Tags: combinatorics , irrational , n-tuples

An $n$-tuple $(x_1, x_2,\dots, x_n)$ is called unearthly if $q_1x_1 +q_2x_2 +\dots+q_nx_n$ is irrational for any non-negative rational coefficients $q_1, q_2, \dots, q_n$ where $q_i$’s are not all zero. Prove that it is possible to select an unearthly $n$-tuple from any $2n-1$ distinct irrational numbers.

1999 Kazakhstan National Olympiad, 1

Tags: algebra , irrational number , irrational

Prove that for any real numbers $ a_1, a_2, \dots, a_ {100} $ there exists a real number $ b $ such that all numbers $ a_i + b $ ($ 1 \leq i \leq 100 $) are irrational.