Olympiad problems

1986 All Soviet Union Mathematical Olympiad, 437

Prove that the sum of all numbers representable as $\frac{1}{mn}$, where $m,n$ -- natural numbers, $1 \le m < n \le1986$, is not an integer.

1918 Eotvos Mathematical Competition, 2

Tags: integer , reciprocal , number theory

Find three distinct natural numbers such that the sum of their reciprocals is an integer.

1986 Tournament Of Towns, (122) 4

Tags: sum , reciprocal , product , algebra

Consider subsets of the set $1 , 2,..., N$. For each such subset we can compute the product of the reciprocals of each member. Find the sum of all such products.

2003 Estonia National Olympiad, 4

Tags: divisor , sum , reciprocal , number theory

Call a positive integer [i]lonely [/i] if the sum of reciprocals of its divisors (including $1$ and the integer itself) is not equal to the sum of reciprocals of divisors of any other positive integer. Prove that a) all primes are lonely, b) there exist infinitely many non-lonely positive integers.

2013 NZMOC Camp Selection Problems, 3

Tags: algebra , number theory , reciprocal

Prove that for any positive integer $n > 2$ we can find $n$ distinct positive integers, the sum of whose reciprocals is equal to $1$.

1951 Moscow Mathematical Olympiad, 204

Tags: number theory , least common multiple , reciprocal , sum

* Given several numbers each of which is less than $1951$ and the least common multiple of any two of which is greater than $1951$. Prove that the sum of their reciprocals is less than $2$.

1948 Moscow Mathematical Olympiad, 141

Tags: number theory , sum , reciprocal

The sum of the reciprocals of three positive integers is equal to $1$. What are all the possible such triples?

2024 Bulgaria National Olympiad, 3

Tags: functional equation , algebra , reciprocal , bijectivity

Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb{R}^{+}$, such that $$f(af(b)+a)(f(bf(a))+a)=1$$ for any positive reals $a, b$.

2018 Saudi Arabia GMO TST, 1

Tags: algebra , sequence , inequalities , recurrence relation , reciprocal

Let $\{x_n\}$ be a sequence defined by $x_1 = 2$ and $x_{n+1} = x_n^2 - x_n + 1$ for $n \ge 1$. Prove that $$1 -\frac{1}{2^{2^{n-1}}} < \frac{1}{x_1}+\frac{1}{x_2}+ ... +\frac{1}{x_n}< 1 -\frac{1}{2^{2^n}}$$ for all $n$

2022 Mexico National Olympiad, 1

Tags: prime , reciprocal

A number $x$ is "Tlahuica" if there exist prime numbers $p_1,\ p_2,\ \dots,\ p_k$ such that \[x=\frac{1}{p_1}+\frac{1}{p_2}+\dots+\frac{1}{p_k}.\] Find the largest Tlahuica number $x$ such that $0<x<1$ and there exists a positive integer $m\leq 2022$ such that $mx$ is an integer.