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2016/17 AHSMC Part 1 Problem 4

Problem 4 of 16

A quadratic polynomial f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb and cc are integers, satisfies f(2)=4f(2) = 4 and f(3)=9f(3) = 9. The number of such polynomials is:

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Solution

Using the given condition we get 4a+2b+c=44a + 2b + c = 4, 9a+3b+c=99a + 3b + c = 9 from which, solving for bb, cc in terms of aa, one obtains b=5(1−a)b = 5(1-a), c=6(a−1)c = 6(a-1), that is, infinitely many integer solutions. The answer is (E).

University of Alberta