AMC 10 · 2020 · #14

Grade 7 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

vieta-formulaspolynomial-factoringsymmetric-polynomialsfraction-arithmetic identify-subproblemsconvert-to-algebra ↑ Prerequisites: vieta-formulaspolynomial-factoring

📏 Long solution 💡 3 insights

Problem

Real numbers x and y satisfy x + y = 4 and xy = -2. Compute the value of x +

\frac{x³}{y²}

\frac{y³}{x²}

+ y.

Pick an answer.

(A)

360

(B)

400

(C)

420

(D)

440

(E)

480

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Tool #7 (Subproblems): the target E has four terms that don't obviously combine. Group them as (x + $\frac{y³}{x²}$ ) + ( $\frac{x³}{y²}$ + y) = $\frac{x³ + y³}{x²}$ + $\frac{x³ + y³}{y²}$ , then factor out x³ + y³ to get E = (x³ + y³)( $\frac{1}{x²}$ + $\frac{1}{y²}$ ) = $\frac{(x³ + y³)(x² + y²)}{x² y²}$ . Tool #13 (Algebra): compute the symmetric sums x² + y², x³ + y³, x² y² from x + y = 4, xy = -2. Tool #3 (Eliminate): match the final number against the five choices.

1STEP 1

Rewrite x = $\frac{x³}{x²}$ and y = $\frac{y³}{y²}$ so all four terms share denominators, then regroup: E = $\frac{x³ + y³}{x²}$ + $\frac{x³ + y³}{y²}$ .

E =

\frac{x³ + y³}{x²}

\frac{x³ + y³}{y²}

2STEP 2

Factor out the common x³ + y³ and combine the fractions over x² y²: E = $\frac{(x³ + y³)(x² + y²)}{x² y²}$ .

E =

\frac{(x³ + y³)(x² + y²)}{x² y²}

3STEP 3

From (x + y)² = x² + 2xy + y²: x² + y² = 4² - 2(-2) = 16 + 4 = 20.

x² + y² = (x+y)² - 2xy = 16 + 4 = 20

4STEP 4

From (x+y)³ = (x³ + y³) + 3xy(x+y): x³ + y³ = 4³ - 3(-2)(4) = 64 + 24 = 88.

x³ + y³ = (x+y)³ - 3xy(x+y) = 64 + 24 = 88

5STEP 5

Square the product: x² y² = (xy)² = (-2)² = 4.

x² y² = (xy)² = 4

6STEP 6

Substitute the three pieces: E = $\frac{88 · 20}{4}$ = $\frac{1760}{4}$ = 440.

E =

\frac{88 · 20}{4}

= 22 · 20 = 440

7STEP 7

Only choice (D) equals 440; dropping the x² + y² factor or a sign slip gives the smaller options.

E = 440 → (D)

Answer

440

The two roots of t² - 4t - 2 = 0 are x = 2 + √(6), y = 2 - √(6) (since x + y = 4, xy = -2). Then x ≈ 4.449, y ≈ -0.449, so

\frac{x³}{y²}

≈

\frac{88.0}{0.202}

≈ 436 and

\frac{y³}{x²}

≈

\frac{-0.091}{19.79}

≈ -0.005. Adding x + y = 4 gives E ≈ 440. ✓ Matches.

💡Key takeaway

This AMC 10 problem only needs Grade 7 algebra you already know! Rewrite x = $\frac{x³}{x²}$ and y = $\frac{y³}{y²}$ so the four terms factor as E = $\frac{(x³ + y³)(x² + y²)}{x² y²}$ . The identities x² + y² = (x+y)² - 2xy = 20 and x³ + y³ = (x+y)³ - 3xy(x+y) = 88 and x² y² = 4 are all you need. Plug in: E = $\frac{88 · 20}{4}$ = 440, answer (D).

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