Problem
ALG-B1-M05-P026 Eight Consecutive Integers
#26
★★★★★ Level 5 of 5
Eight consecutive positive integers are given. Prove that six of them can be labelled \(a,b,c,d,e,f\) so that \(\frac a{b+c}+\frac d{e+f}=1\).
Inspired by regional olympiad method · 2026 · Grade 11 · Problem 1
Take the first six numbers and apply the symmetric construction.
Among the given numbers are \(n,n+1,\ldots,n+7\). Take the first six: \(n,\ldots,n+5\).
As in the previous construction, set \(a=n\), \(d=n+5\), \(b=n+1\), \(c=n+4\), \(e=n+2\), \(f=n+3\). Then both denominators equal \(2n+5\), and \(a+d=2n+5\). Hence the sum is \(1\).
The same architecture, without directly duplicating the source statement.