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Found problems: 2

2024 Bosnia and Herzegovina Junior BMO TST, 2.
Tags: number thoery , modular arithmetic
Determine all $x$, $y$, $k$ and $n$ positive integers such that: $10^x$ + $10^y$ + $n!$ = $2024^k$

VI Soros Olympiad 1999 - 2000 (Russia), 9.5
Tags: number thoery , algebra , Sequence
Let b be a given real number. The sequence of integers $a_1, a_2,a_3, ...$ is such that $a_1 =(b]$ and $a_{n+1}=(a_n+b]$ for all $n\ge 1$ Prove that the sum $a_1+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{n}$ is an integer number for any natural $n$ . (In the condition of the problem, $(x]$ denotes the smallest integer that is greater than or equal to $x$)