2020 AMC 10A Problem 14

Name: 2020 AMC 10A Problem 14 — Video Solution
Uploaded: 2022-11-05T14:22:22-07:00
Duration: 2 h 28 min 50 s
Description: Po-Shen Loh presents the video solution to 2020 AMC 10A Problem 14, from "2020 AMC 10A LIVE Solve".

Question

Real numbers

x

and

y

satisfy

x + y = 4

and

x \cdot y = -2.

What is the value of

x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?

360

400

420

440

480

Answer: D

Concepts:Newton’s Sums Vieta’s Formulas

Difficulty rating: 1480

Small hint:

Use power sums

S_k=x^k+y^k

Big hint:

The relation

t^2-4t-2=0

gives a recurrence for

S_k

Video solution:

Click to load, then click again to play

Written solution:

Let

S_k=x^k+y^k

. Since

x+y=4

and

xy=-2

, the numbers

x

and

y

satisfy

t^2-4t-2=0

, so

S_k=4S_{k-1}+2S_{k-2}

.

Using

S_0=2

and

S_1=4

, we get

S_2=20

,

S_3=88

,

S_4=392

, and

S_5=1744

. The expression is

x+y+\dfrac{x^5+y^5}{x^2y^2}

=4+\dfrac{1744}{4}=440

. Thus, D is the correct answer.