Skip to main content

2024/25 AHSMC Part 1 Problem 12

Problem 12 of 16

Let f(x)=x2+nxf(x) = x^2 + nx, where nn is an integer. The number of values of nn for which the equations f(x)=0f(x) = 0 and f(f(x))=0f(f(x)) = 0 have the same set of real solutions (not considering multiplicities) is:

Answer choices

Show solution

Solution

We have f(f(x))=f(x)(f(x)+n)=f(x)(x2+nx+n)f(f(x)) = f(x)\bigl(f(x)+n\bigr) = f(x)(x^2 + nx + n). The two equations have the same real roots if and only if x2+nx+n=0x^2 + nx + n = 0 has no real solution or the same solutions as x2+nx=0x^2 + nx = 0. This happens if and only if n2−4n<0n^2 - 4n < 0 or n=0n = 0, i.e. n∈{0,1,2,3}n \in \{0,1,2,3\}. The answer is (D).

University of Alberta