Problem A. 463. (October 2008)
A. 463. Let a1<a2<...<an and b1<b2<...<bn be real numbers. Show that
Deadline expired on 17 November 2008.
Solution. Apply induction on n. For n=1 the statement is ea1b1>0 which is obvious. Now suppose n>1 and assume that the statement is true for all smaller values.
Let ci=ai-a1>0. Then
so it is sufficient to prove that the last determinant is positive.
To eliminate the first row, subtract the (n-1)th column from the nth column. Then subtract the (n-2)th column from the (n-1)th column, and so on, finally subtract the first column from the second column. Then
Consider the function
By Lagrange's mean value theorem, there exists a b1<x1<b2 such that f(b2)-f(b1)=(b2-b1)f'(x1), i.e.
Repeating the same argument for each column, it can be btained that there exist real numbers xi(bi,bi+1) (1in-1) such that
By the induction hypothesis, this is positive.