2025 AMC 12A Problema 19

Question

Sean

a,

b,

y

c

las raíces del polinomio

x^3 + kx + 1.

¿Cuánto vale la suma

\begin{aligned} &a^3b^2 + a^2b^3 + b^3c^2 \\ &\quad {}+ b^2c^3 + c^3a^2 + c^2a^3? \end{aligned}

Let

a,

b,

and

c

be the roots of the polynomial

x^3 + kx + 1.

What is the sum

\begin{aligned} &a^3b^2 + a^2b^3 + b^3c^2 \\ &\quad {}+ b^2c^3 + c^3a^2 + c^2a^3? \end{aligned}

-k

-k + 1

1

k - 1

k

Por Vieta,

a + b + c = 0,

ab + bc + ca = k,

y

abc = -1

By Vieta,

a + b + c = 0,

ab + bc + ca = k,

and

abc = -1

Agrupa como

a^2b^2(a + b) + b^2c^2(b + c)

+ c^2a^2(c + a)

y reemplaza

a + b = -c

Group as

a^2b^2(a + b) + b^2c^2(b + c)

+ c^2a^2(c + a)

and replace

a + b = -c

Por las fórmulas de Vieta,

a + b + c = 0,

ab + bc + ca = k,

y

abc = -1.

Agrupa la suma como

\begin{aligned} &a^2b^2(a + b) + b^2c^2(b + c) \\ &\quad {}+ c^2a^2(c + a). \end{aligned}

Como

a + b + c = 0,

tenemos

a + b = -c,

b + c = -a,

c + a = -b.

Así que la suma es igual a

\begin{gathered} -a^2b^2 c - ab^2c^2 - a^2bc^2 \\ = -abc(ab + bc + ca) \\ = -(-1)(k) = k. \end{gathered}

Por lo tanto, la respuesta correcta es E.

By Vieta’s formulas,

a + b + c = 0,

ab + bc + ca = k,

and

abc = -1.

Group the sum as

\begin{aligned} &a^2b^2(a + b) + b^2c^2(b + c) \\ &\quad {}+ c^2a^2(c + a). \end{aligned}

Since

a + b + c = 0,

we have

a + b = -c,

b + c = -a,

c + a = -b.

So the sum equals

\begin{gathered} -a^2b^2 c - ab^2c^2 - a^2bc^2 \\ = -abc(ab + bc + ca) \\ = -(-1)(k) = k. \end{gathered}

Thus, the correct answer is E.

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