2025 AMC 10A Problem 9

The graph of $y = f(x-a)$ passes through $(1,25)$ exactly when $f(1 - a) = 25$

Factor $f(x) = 100x(x-1)(x-2)$ and count solutions of $f(t) = 25$; note that $f(0.5)$ exceeds $25$

The graph passes through $(1,25)$ exactly when $f(1 - a) = 25.$ Let $t = 1 - a,$ so we count solutions of $f(t) = 25.$ Factor $f(x) = 100x(x-1)(x-2),$ with roots $0, 1, 2.$ On $(0,1),$ the function is positive and $f(0.5) = 37.5 \gt 25,$ so continuity gives one root on each side of $0.5.$ On $(1,2)$ the function is negative, while for $x \gt 2$ it increases from $0$ to infinity, giving one more root. A cubic equation has at most $3$ real roots, so these are all the solutions. Each gives one $a,$ so there are $3$ values. Thus, C is the correct answer.