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2025/26 AHSMC Part 1 Problem 8

Problem 8 of 16

For how many integers n≥1n \ge 1 do there exist real numbers aa, bb such that a+b=nanda2+b2=25? a + b = n \qquad\text{and}\qquad a^2 + b^2 = 25?

Answer choices

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Solution

Such a,ba,b exist precisely when x2+(n−x)2=25x^2 + (n-x)^2 = 25 has a real solution, i.e. (2x−n)2=50−n2(2x-n)^2 = 50 - n^2. This requires 50−n2≥050-n^2 \ge 0. For integers n≥1n \ge 1 there are 77 possibilities, namely n=1,…,7n = 1,\ldots,7. The answer is (E).

University of Alberta