Olympiad problems

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Found problems: 3

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ICMC 4, 5

Find all composite positive integers $m$ such that, whenever the product of two positive integers $a$ and $b$ is $m$, their sum is a power of $2$. [i]Proposed by Harun Khan[/i]

2020 Brazil National Olympiad, 3

Tags: powers of 2

Consider an inifinte sequence $x_1, x_2,\dots$ of positive integers such that, for every integer $n\geq 1$: [list] [*]If $x_n$ is even, $x_{n+1}=\dfrac{x_n}{2}$; [*]If $x_n$ is odd, $x_{n+1}=\dfrac{x_n-1}{2}+2^{k-1}$, where $2^{k-1}\leq x_n<2^k$.[/list] Determine the smaller possible value of $x_1$ for which $2020$ is in the sequence.

2024 Czech-Polish-Slovak Junior Match, 5

Tags: palindrome , digit , powers of 2 , number theory

Is there a positive integer $n$ such that when we write the decimal digits of $2^n$ in opposite order, we get another integer power of $2$?

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