Olympiad problems

1987 Bundeswettbewerb Mathematik, 4

Let $1<k\leq n$ be positive integers and $x_1 , x_2 , \ldots , x_k$ be positive real numbers such that $x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.$ a) Show that $x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.$ b) Find all numbers $k,n$ and $x_1, x_2 ,\ldots , x_k$ for which equality holds.

2023 Ukraine National Mathematical Olympiad, 8.5

Tags: equality , algebra

Do there exist $10$ real numbers, not all of which are equal, each of which is equal to the square of the sum of the remaining $9$ numbers? [i]Proposed by Bogdan Rublov[/i]

2023 Ukraine National Mathematical Olympiad, 9.1

Tags: combinatorics , equality

$n \ge 4$ real numbers are arranged in a circle. It turned out that for any four consecutive numbers $a, b, c, d$, that lie on the circle in this order, holds $a+d = b+c$. For which $n$ does it follow that all numbers on the circle are equal? [i]Proposed by Oleksiy Masalitin[/i]

1994 All-Russian Olympiad, 5

Tags: equality , algebra

Prove the equality $$\frac{a_1}{a_2(a_1+a_2)}+\frac{a_2}{a_3(a_2+a_3)}+...+\frac{a_n}{a_1(a_n+a_1)}=\frac{a_2}{a_1(a_1+a_2)}+\frac{a_3}{a_2(a_2+a_3)}+...+\frac{a_1}{a_n(a_n+a_1)} $$ (R. Zhenodarov)

2017 Bosnia and Herzegovina EGMO TST, 4

Tags: algebra , triplets , equality , inequality

Let $a$, $b$, $c$, $d$ and $e$ be distinct positive real numbers such that $a^2+b^2+c^2+d^2+e^2=ab+ac+ad+ae+bc+bd+be+cd+ce+de$ $a)$ Prove that among these $5$ numbers there exists triplet such that they cannot be sides of a triangle $b)$ Prove that, for $a)$, there exists at least $6$ different triplets