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2012/13 AHSMC Part 1 Problem 10

Problem 10 of 16

The positive integer nn is such that between n2+1n^2 + 1 and 2n22n^2 there are exactly five different perfect squares. How many such nn can we find?

Answer choices

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Solution

Solving (n+5)2<2n2<(n+6)2(n + 5)^2 < 2n^2 < (n + 6)^2 yields 50<(n−5)250 < (n - 5)^2 and (n−6)2<72(n - 6)^2 < 72. Thus 8≤n−58 \le n - 5 and n−6<9n - 6 < 9 or 13≤n≤1413 \le n \le 14. The answer is (C).

University of Alberta