IMO 1984 SL 19

Origin: CAN

Problem

The triangular array (an,k) of numbers is given by an,1 = 1/n, for n = 1, 2, . . ., an,k+1 = an−1,k −an,k, for 1 \leqk \leqn −1. Find the harmonic mean of the 1985th row.

Solution

First, we shall prove that the numbers in the nth row are exactly the numbers n
n−1  , n
n−1  , n
n−1  , . . . , n
n−1 n−1 . (1) The proof of this fact can be done by induction. For small n, the statement can be easily verified. Assuming that the statement is true for some n, we have that the kth element in the (n + 1)st row is, as is directly verified, n
n−1 k−1  − (n + 1)
n k−1  = (n + 1)
n k . Thus (1) is proved. Now the geometric mean of the elements of the nth row becomes: n n
n−1  \cdot
n−1  \cdot \cdot \cdot
n−1 n−1  \geq n  (n−1 0 )+(n−1 1 )+\cdot\cdot\cdot+(n−1 n−1) n  = 2n−1 . The desired result follows directly from substituting n = 1984.