Olympiad problems

2012 Belarus Team Selection Test, 1

A cubic trinomial $x^3 + px + q$ with integer coefficients $p$ and $q$ is said to be [i]irrational [/i] if it has three pairwise distinct real irrational roots $a_1,a_2, a_3$ Find all irrational cubic trinomials for which the value of $|a_1| + [a_2| + |a_3|$ is the minimal possible. (E. Barabanov)

min polynomial cubic sum cubic equation irrational algebra polynomials

1999 Kazakhstan National Olympiad, 1

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.

irrational number irrational algebra modular arithmetic

1974 Putnam, B3

Prove that if $a$ is a real number such that $$\cos \pi a= \frac{1}{3},$$ then $a$ is irrational.

cosine irrational algebra

2012 Thailand Mathematical Olympiad, 10

Let $x$ be an irrational number. Show that there are integers $m$ and $n$ such that $\frac{1}{2555}< mx + n <\frac{1}{2012}$

inequalities number theory irrational algebra

2002 Junior Balkan Team Selection Tests - Romania, 1

Let $a$ be an integer. Prove that for any real number $x, x^3 < 3$, both the numbers $\sqrt{3 -x^2}$ and $\sqrt{a - x^3}$ cannot be rational.

radical rational irrational algebra inequalities