Olympiad problems

2001 Estonia National Olympiad, 4

We call a triple of positive integers $(a, b, c)$ [i]harmonic [/i] if $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$. Prove that, for any given positive integer $c$, the number of harmonic triples $(a, b, c)$ is equal to the number of positive divisors of $c^2$.

2021 China Team Selection Test, 2

Tags: harmonic , geometry

Let triangle$ABC(AB<AC)$ with incenter $I$ circumscribed in $\odot O$. Let $M,N$ be midpoint of arc $\widehat{BAC}$ and $\widehat{BC}$, respectively. $D$ lies on $\odot O$ so that $AD//BC$, and $E$ is tangency point of $A$-excircle of $\bigtriangleup ABC$. Point $F$ is in $\bigtriangleup ABC$ so that $FI//BC$ and $\angle BAF=\angle EAC$. Extend $NF$ to meet $\odot O$ at $G$, and extend $AG$ to meet line $IF$ at L. Let line $AF$ and $DI$ meet at $K$. Proof that $ML\bot NK$.

2020 Iran Team Selection Test, 4

Tags: circumcircle , geometry , harmonic , homothety , incenter

Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

1970 Czech and Slovak Olympiad III A, 1

Tags: number theory , prime , fraction , harmonic , number , prime divisor

Let $p>2$ be a prime and $a,b$ positive integers such that \[\frac ab=1+\frac12+\frac13+\cdots+\frac{1}{p-1}.\] Show that $p$ is a divisor of $a.$

1968 Kurschak Competition, 1

Tags: algebra , harmonic

In an infinite sequence of positive integers every element (starting with the second) is the harmonic mean of its neighbors. Show that all the numbers must be equal.

2021 China Team Selection Test, 2

Tags: geometry , harmonic

Let triangle$ABC(AB<AC)$ with incenter $I$ circumscribed in $\odot O$. Let $M,N$ be midpoint of arc $\widehat{BAC}$ and $\widehat{BC}$, respectively. $D$ lies on $\odot O$ so that $AD//BC$, and $E$ is tangency point of $A$-excircle of $\bigtriangleup ABC$. Point $F$ is in $\bigtriangleup ABC$ so that $FI//BC$ and $\angle BAF=\angle EAC$. Extend $NF$ to meet $\odot O$ at $G$, and extend $AG$ to meet line $IF$ at L. Let line $AF$ and $DI$ meet at $K$. Proof that $ML\bot NK$.

2020 Iran Team Selection Test, 4

Tags: circumcircle , geometry , incenter , harmonic , homothety

Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]