Olympiad problems

2018 OMMock - Mexico National Olympiad Mock Exam, 3

Find all $n$-tuples of real numbers $(x_1, x_2, \dots, x_n)$ such that, for every index $k$ with $1\leq k\leq n$, the following holds: \[ x_k^2=\sum\limits_{\substack{i < j \\ i, j\neq k}} x_ix_j \] [i]Proposed by Oriol Solé[/i]

2025 Kyiv City MO Round 1, Problem 4

Tags: algebra , algebraic manipulations , value

Distinct real numbers $ a, b, c $ satisfy the following condition: \[ \frac{a - b}{a^3b^3} + \frac{b - c}{b^3c^3} + \frac{c - a}{c^3a^3} = 0. \] What are the possible values of the expression \[ \frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2}? \] [i]Proposed by Vadym Solomka[/i]

2022 Irish Math Olympiad, 6

Tags: algebra , inequalities , algebraic manipulations

6. Suppose [i]a[/i], [i]b[/i], [i]c[/i] are real numbers such that [i]a[/i] + [i]b[/i] + [i]c[/i] = 1. Prove that \[a^3 + b^3 + c^3 + 3(1-a)(1-b)(1-c) = 1.\]

2023 Bulgaria JBMO TST, 1

Tags: algebra , system of equations , algebraic manipulations

Determine all triples $(x,y,z)$ of real numbers such that $x^4 + y^3z = zx$, $y^4 + z^3x = xy$ and $z^4 + x^3y = yz$.