is given. prove that there is 3 numbers a, b, c that: $0.5<$a^2/bc<2 using pigeonhole principle.(n>2)

Question

$2n-1$ numbers from {${1, 2, 3, ...,2^n-2}$} is given. prove that there is 3 numbers a, b, c that: $0.5<$a^2/bc<2 using pigeonhole principle.(n>2) we do not know which numbers selected. a, b, c are distinct numbers.

Answer 1

For $k\in\{1,...,n-1\}$, let $X_k=\{2^k-1,...,2^{k+1}-2\}$.

Every element of $\{1,...,2^n-2\}$ lies in exactly one of the sets $X_1,...,X_{n-1}$, hence by the pigeonhole principle, of the $2n-1$ selected numbers, at least one of those sets, $X_k$ say, contains at least $3$ of the selected numbers.

Thus suppose $X_k$ contains the $3$ selected numbers $a,b,c$ with $b < a < c$.$\;$Then $$ \frac{1}{2} = \frac{2^k-1}{2^{k+1}-2} \le \frac{b}{c} = \frac{b^2}{bc} < \frac{a^2}{bc} < \frac{c^2}{bc} = \frac{c}{b} \le \frac{2^{k+1}-2}{2^k-1} = 2 $$