IndeterminateEquation IMO

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Solve in positive integers $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{4}{5}$

Find all integers $a$, $b$, $c$ with $1 < a < b < c$ such that the number $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$.

Prove that if positive integer $a$ and $b$ are such that $ab+1$ divides $a^2 + b^2$. then $$\frac{a^2+b^2}{ab+1}$$ is a square number.

Find all postive integers $(a,b,c)$ such that $$ab-c,\quad bc-a,\quad ca-b$$are all powers of $2$. Proposed by Serbia

In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $m − 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n^{th}$ and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?