Olympiad problems

2025 Kosovo National Mathematical Olympiad`, P2

Find the smallest natural number $k$ such that the system of equations $$x+y+z=x^2+y^2+z^2=\dots=x^k+y^k+z^k $$ has only one solution for positive real numbers $x$, $y$ and $z$.

2011 Germany Team Selection Test, 1

A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$ [i]Proposed by Gerhard Wöginger, Austria[/i]

get the smallest sequence inequality algebra casework inequalities mathematical induction modular arithmetic parity proof by contradiction sequences

2010 IMO Shortlist, 4

A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$ [i]Proposed by Gerhard Wöginger, Austria[/i]

get the smallest sequence inequality algebra casework inequalities mathematical induction modular arithmetic parity proof by contradiction sequences

2011 Germany Team Selection Test, 1

A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$ [i]Proposed by Gerhard Wöginger, Austria[/i]

get the smallest sequence inequality algebra casework inequalities mathematical induction modular arithmetic parity proof by contradiction sequences