AMC 10 · 2015 · #16

Grade 8 arithmetic

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

systems-of-equationssymmetric-polynomialsdifference-of-squares symmetry-argument ↑ Prerequisites: systems-of-equations

📏 Medium solution 💡 3 insights

Problem

Two numbers x and y are tied together by a pair of equations: y plus 4 equals (x minus 2) squared, and x plus 4 equals (y minus 2) squared. The two numbers are not equal. Find the value of x squared plus y squared.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Organize Information in More Ways

Solving for x and y one at a time leads to an ugly quartic. The two equations are mirror images, so instead reorganize them: add them to get one symmetric relation and subtract them to get another. Adding produces a link between x² + y² and x + y; subtracting, with a difference-of-squares factor, hands over x + y directly. We never need x and y separately, only the combinations the question actually asks about.

1STEP 1

Add the two equations

Adding the two equations cancels the constant 8, linking x² + y² directly to x + y.

(y+4)+(x+4) = (x-2)²+(y-2)² → x+y+8 = x²+y²-4(x+y)+8 → x²+y² = 5(x+y)

2STEP 2

Subtract the two equations

Subtracting the equations gives y - x on the left and a difference of squares on the right.

(y+4)-(x+4) = (x-2)²-(y-2)² → y-x = (x+y-4)(x-y)

3STEP 3

Cancel the x - y factor

Since x ≠ y, the shared (x - y) factor is nonzero, so dividing it out leaves a simple equation for x + y.

-(x-y) = (x+y-4)(x-y) → -1 = x+y-4 → x+y = 3

4STEP 4

Combine to get x² + y²

Step 1 linked x² + y² to x + y; step 3 gave x + y = 3, so plug it in for the answer.

x²+y² = 5(x+y) = 5· 3 = 15

Answer

Cross-check with the individual numbers. From x + y = 3 and x² + y² = 15, the product is xy = ((x+y)² - (x²+y²))/2 = (9 - 15)/2 = -3, so x and y are the two roots of t² - 3t - 3 = 0, namely (3 ± √(21))/2. These are real and unequal, matching the condition that x differs from y. Squaring and adding them recovers (x+y)² - 2xy = 9 - 2(-3) = 15, the same value, so the shortcut and the brute-force route agree.

💡Key takeaway

When two equations are mirror images, add and subtract them — the sum and difference often hand you the combination you need.

Add the two equations
Subtract the two equations
Cancel the x - y factor
Combine to get x² + y²

Add the two equations

Subtract the two equations

Cancel the x - y factor

Combine to get x² + y²

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