Olympiad problems

2018 Bosnia And Herzegovina - Regional Olympiad, 1

if $a$, $b$ and $c$ are real numbers such that $(a-b)(b-c)(c-a) \neq 0$, prove the equality: $\frac{b^2c^2}{(a-b)(a-c)}+\frac{c^2a^2}{(b-c)(b-a)}+\frac{a^2b^2}{(c-a)(c-b)}=ab+bc+ca$

2018 VJIMC, 1

Tags: equation , power , real number

Find all real solutions of the equation \[17^x+2^x=11^x+2^{3x}.\]

2024 Brazil National Olympiad, 5

Tags: functional equation , real number , algebra

Let $ \mathbb{R} $ be the set of real numbers. Determine all functions $ f: \mathbb{R} \to \mathbb{R} $ such that, for any real numbers $ x $ and $ y $, \[ f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1. \]

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , equation , real number

Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$ $$y=\frac{2x^2}{1+x^2}$$ $$z=\frac{2y^2}{1+y^2}$$

2025 India National Olympiad, P4

Tags: inequalities , real number

Let $n\ge 3$ be a positive integer. Find the largest real number $t_n$ as a function of $n$ such that the inequality \[\max\left(|a_1+a_2|, |a_2+a_3|, \dots ,|a_{n-1}+a_{n}| , |a_n+a_1|\right) \ge t_n \cdot \max(|a_1|,|a_2|, \dots ,|a_n|)\] holds for all real numbers $a_1, a_2, \dots , a_n$ . [i]Proposed by Rohan Goyal and Rijul Saini[/i]

2015 Germany Team Selection Test, 1

Tags: root , algebra , polynomial , real number , integer

Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

1994 Bundeswettbewerb Mathematik, 1

Tags: pigeonhole , decimal representation , real number , digit , algebra

Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.

2020 Dürer Math Competition (First Round), P3

Tags: real number , counting , extremal combinatorics , algebra

At least how many non-zero real numbers do we have to select such that every one of them can be written as a sum of $2019$ other selected numbers and a) the selected numbers are not necessarily different? b) the selected numbers are pairwise different?

2003 Bosnia and Herzegovina Team Selection Test, 6

Tags: real number , algebra

Let $a$, $b$ and $c$ be real numbers such that $\mid a \mid >2$ and $a^2+b^2+c^2=abc+4$. Prove that numbers $x$ and $y$ exist such that $a=x+\frac{1}{x}$, $b=y+\frac{1}{y}$ and $c=xy+\frac{1}{xy}$.