2025 AMC 10A Problem 9

Problem 9 of 25EasierAlgebra

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)?

Show solution

Solution

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.

(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.

(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.