Track / Stage 5 / 335 of 400 #935 of 1964

Problem 935

AIME late

Number theory Difficulty 5.8 Prove it

Sis: *Take $n(\geqslant 2)$ distinct fractions in the interval $(0,1)$ . Prove: the sum of the denominators of these fractions is not less than $\frac{1}{3} n^{\frac{3}{2}}$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Let the $n$ fractions taken be $\frac{a_{1}}{b_{1}}t} 1 \leqslant \frac{1}{t} \sum_{b,>t} b_{i} \leqslant \frac{B}{t}$ , so
$n=\sum_{b_{1}1} 1 \leqslant t^{2}+\frac{B}{t} .$

Taking $t=B^{\frac{1}{3}}$ (to make the two terms on the right side of (1) equal), then $2 B^{\frac{2}{3}} \geqslant n$ , thus
$B \geqslant\left(\frac{n}{2}\right)^{3 / 2}>\frac{1}{3} n^{3 / 2} .$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.