2025 AMC 10A Problems/Problem 9

Let $f(x) = 100x^3 - 300x^2 + 200x$ . For how many real numbers $a$ does the graph of $y = f(x - a)$ pass through the point $(1, 25)$ ?

(A) $1$ (B) $2$ (C) $3$ (D) $4$ (E) more than $4$

Solution

The problem boils down to how many real roots does the equation $100(1-a)^3 - 300(1-a)^2 + 200(1-a) = 25$ have? Using Descarte's Rule of Signs we find there shouldn't be any imaginary roots, so the answer is $\boxed{3}$ real roots.

~grogg007

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2025 AMC 10A Problems/Problem 9

Solution