Problem A. 463. (October 2008)
A. 463. Let a_{1}<a_{2}<...<a_{n} and b_{1}<b_{2}<...<b_{n} be real numbers. Show that
(5 pont)
Deadline expired on 17 November 2008.
Solution. Apply induction on n. For n=1 the statement is e^{a1b1}>0 which is obvious. Now suppose n>1 and assume that the statement is true for all smaller values.
Let c_{i}=a_{i}a_{1}>0. Then
so it is sufficient to prove that the last determinant is positive.
To eliminate the first row, subtract the (n1)th column from the nth column. Then subtract the (n2)th column from the (n1)th column, and so on, finally subtract the first column from the second column. Then
Consider the function
Then
By Lagrange's mean value theorem, there exists a b_{1}<x_{1}<b_{2} such that f(b_{2})f(b_{1})=(b_{2}b_{1})f'(x_{1}), i.e.
Repeating the same argument for each column, it can be btained that there exist real numbers x_{i}(b_{i},b_{i+1}) (1in1) such that
By the induction hypothesis, this is positive.
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