Found problems: 10
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.
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.
Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb{R}^{+}$, such that $$f(af(b)+a)(f(bf(a))+a)=1$$ for any positive reals $a, b$.
* 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 $2$.
The sum of the reciprocals of three positive integers is equal to $1$. What are all the possible such triples?
Prove that for any positive integer $n > 2$ we can find $n$ distinct positive integers, the sum of whose reciprocals is equal to $1$.
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.
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.
Find three distinct natural numbers such that the sum of their reciprocals is an integer.
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 $n$