Olympiad problems

2014 IMO, 1

Tags: IMO , algebra , inequalities , Sequence , IMO 2014 , IMO P1 , Unique

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]

2019 IMO, 1

Tags: functional equation , IMO Shortlist , IMO 2019 , IMO P1

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$ [i]Proposed by Liam Baker, South Africa[/i]

2014 Contests, 1

Tags: IMO , algebra , inequalities , Sequence , IMO 2014 , IMO P1 , Unique

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]

2018 IMO, 1

Tags: geometry , IMO , imo 2018 , IMO P1 , 2018

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line. [i]Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece[/i]

2019 IMO Shortlist, A1

Tags: functional equation , IMO Shortlist , IMO 2019 , IMO P1

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$ [i]Proposed by Liam Baker, South Africa[/i]

2018 IMO Shortlist, G1

Tags: geometry , IMO , imo 2018 , IMO P1 , 2018

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line. [i]Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece[/i]

2014 IMO Shortlist, A1

Tags: IMO , algebra , inequalities , Sequence , IMO 2014 , IMO P1 , Unique

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]