Olympiad problems

2017 South Africa National Olympiad, 3

A representation of $\frac{17}{20}$ as a sum of reciprocals $$ \frac{17}{20} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} $$ is called a [i]calm representation[/i] with $k$ terms if the $a_i$ are distinct positive integers and at most one of them is not a power of two. (a) Find the smallest value of $k$ for which $\frac{17}{20}$ has a calm representation with $k$ terms. (b) Prove that there are infinitely many calm representations of $\frac{17}{20}$.

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1989 Bundeswettbewerb Mathematik, 4

Let $n$ be an odd positive integer. Show that the equation $$ \frac{4}{n} =\frac{1}{x} + \frac{1}{y}$$ has a solution in the positive integers if and only if $n$ has a divisor of the form $4k+3$.

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1992 IMO Longlists, 23

An [i]Egyptian number[/i] is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is $1$. For example, $32 = 2 + 3 + 9 + 18$ is Egyptian because $\frac 12 +\frac 13 +\frac 19 +\frac{1}{18}=1$ . Prove that all integers greater than $23$ are [i]Egyptian[/i].

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2024 ITAMO, 3

A positive integer $n$ is called [i]egyptian[/i] if there exists a strictly increasing sequence $0<a_1<a_2<\dots<a_k=n$ of integers with last term $n$ such that \[\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_k}=1.\] (a) Determine if $n=72$ is egyptian. (b) Determine if $n=71$ is egyptian. (c) Determine if $n=72^{71}$ is egyptian.

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2016 Latvia National Olympiad, 3

Prove that for every integer $n$ ($n > 1$) there exist two positive integers $x$ and $y$ ($x \leq y$) such that $$\frac{1}{n} = \frac{1}{x(x+1)} + \frac{1}{(x+1)(x+2)} + \cdots + \frac{1}{y(y+1)}$$

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