MathDB
Permutation yields equal ratios

Source: Romania JBMO TST 2024 Day 4 P1

July 31, 2024
algebrapermutations

Problem Statement

Let n3n\geqslant 3 be an integer and a1,a2,,ana_1,a_2,\ldots,a_n be pairwise distinct positive real numbers with the property that there exists a permutation b1,b2,,bnb_1,b_2,\ldots,b_n of these numbers such thata1b1=a2b2==an1bn11.\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots=\frac{a_{n-1}}{b_{n-1}}\neq 1.Prove that there exist a,b>0a,b>0 such that {a1,a2,,an}={ab,ab2,,abn}.\{a_1,a_2,\ldots,a_n\}=\{ab,ab^2,\ldots,ab^n\}.
Cristi Săvescu