Olympiad problems

1980 Brazil National Olympiad, 2

Show that for any positive integer $n > 2$ we can find $n$ distinct positive integers such that the sum of their reciprocals is $1$.

2019 Saint Petersburg Mathematical Olympiad, 4

Tags: divisor , reciprocal sum , sum , irreducible , number theory

Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?

2013 Danube Mathematical Competition, 1

Tags: algebra , sum , reciprocal sum

Determine the natural numbers $n\ge 2$ for which exist $x_1,x_2,...,x_n \in R^*$, such that $$x_1+x_2+...+x_n=\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}=0$$

2021 Bundeswettbewerb Mathematik, 2

Tags: reciprocal sum , number theory

The fraction $\frac{3}{10}$ can be written as a sum of two reciprocals in exactly two ways: \[\frac{3}{10}=\frac{1}{5}+\frac{1}{10}=\frac{1}{4}+\frac{1}{20}\] a) In how many ways can $\frac{3}{2021}$ be written as a sum of two reciprocals? b) Is there a positive integer $n$ not divisible by $3$ with the property that $\frac{3}{n}$ can be written as a sum of two reciprocals in exactly $2021$ ways?

2006 China Team Selection Test, 2

Tags: algebra , reciprocal sum , extremal combinatorics , maximization

Given positive integer $n$, find the biggest real number $C$ which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than $1$ is less than $C$, then we can divide the set of numbers into no more than $n$ groups so that the sum of reciprocals of every group is less than $1$.

2010 Ukraine Team Selection Test, 12

Tags: algebra , number theory , rational , sum , reciprocal sum

Is there a positive integer $n$ for which the following holds: for an arbitrary rational $r$ there exists an integer $b$ and non-zero integers $a _1, a_2, ..., a_n$ such that $r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}$ ?

2013 Balkan MO Shortlist, N1

Tags: number theory , prime , reciprocal sum

Let $p$ be a prime number. Determine all triples $(a,b,c)$ of positive integers such that $a + b + c < 2p\sqrt{p}$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{p}$

2006 China Team Selection Test, 2

Tags: algebra , reciprocal sum , extremal combinatorics , maximization

Given positive integer $n$, find the biggest real number $C$ which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than $1$ is less than $C$, then we can divide the set of numbers into no more than $n$ groups so that the sum of reciprocals of every group is less than $1$.

2002 IMO Shortlist, 6

Tags: inequality , reciprocal sum , additive number theory , algebra

Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1,\ldots b_n$ and $c_1,\ldots,c_n$ such that - for each $i$ the set $b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}$ is a subset of $A$, and - the sets $b_iA+c_i$ and $b_jA+c_j$ are disjoint whenever $i\ne j$ Prove that \[{1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.\]

2004 Junior Tuymaada Olympiad, 6

Tags: prime , sum , reciprocal sum , positive integer , number theory

We call a positive integer [i] good[/i] if the sum of the reciprocals of all its natural divisors are integers. Prove that if $ m $ is a [i]good [/i] number, and $ p> m $ is a prime number, then $ pm $ is not [i]good[/i].

1998 Tuymaada Olympiad, 1

Tags: algebra , sum , reciprocal sum

Write the number $\frac{1997}{1998}$ as a sum of different numbers, inverse to naturals.

2024 ITAMO, 3

Tags: egyptian fractions , reciprocal sum , number theory

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.