AMC 10 · 2008 · #5

Grade 7 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

function-evaluationsymmetry-argument convert-to-algebra ↑ Prerequisites: exponents

📏 Medium solution 💡 1 insight

Problem

For real numbers a and b, a new operation is defined by a * b=(a-b)² — subtract the second input from the first, then square the result. For real numbers x and y, the left input is (x-y)² and the right input is (y-x)². What is the value of (x-y)² * (y-x)²?

Pick an answer.

(A)

$\ 0$

(B)

$\ x^2+y^2$

(C)

$\ 2x^2$

(D)

$\ 2y^2$

(E)

$\ 4xy$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The expression looks tangled because the two inputs (x-y)² and (y-x)² are themselves squares. Tool #4 (Introduce a Variable) tames this: name the left input a and the right input b, so the whole problem becomes the clean a * b=(a-b)². The key observation then jumps out — a and b are actually the same number — so the difference a-b is 0. Tool #6 (Guess and Check) confirms the result with concrete numbers, and Tool #3 (Eliminate Possibilities) explains why every non-zero choice (which all still contain x or y) must be wrong once the difference collapses to 0.

1STEP 1

Decode the star rule

The star is not multiplication: name the inputs a=(x-y)² and b=(y-x)², and the task is just a * b=(a-b)².

a=(x-y)², b=(y-x)², a * b=(a-b)²

2STEP 2

The two inputs are equal

The bases are opposites, y-x=-(x-y), and (-k)²=k², so b=(y-x)²=(x-y)²=a — the two inputs are one number.

y-x=-(x-y) → (y-x)²=(x-y)² → a=b

3STEP 3

Test with real numbers

Try x=2, y=1: both inputs come out to 1, and the rule gives 1 * 1=(1-1)²=0.

(2-1)²=1, (1-2)²=1, 1 * 1=(1-1)²=0

4STEP 4

Subtract equal inputs and square

Since a=b the difference is a-b=0, so (a-b)²=0 — x and y vanish, and the answer is choice (A).

a * b=(a-b)²=(a-a)²=0²=0 → (A)

Answer

The whole result rests on one fact: the two inputs (x-y)² and (y-x)² are identical, so their difference is 0 and its square is 0. The numeric test with x=2, y=1 gave 0, and a second test — say x=5, y=3: (5-3)²=4, (3-5)²=4, 4 * 4=(4-4)²=0 — gives 0 again. Because the answer never depends on x or y, every choice that still shows an x or y (B, C, D, E) must be a distractor; only the constant (A) 0 can survive.

💡Key takeaway

When a rule squares the difference of two things and those two things are actually equal, the difference is 0 — so the whole answer is 0.

Decode the star rule
The two inputs are equal
Test with real numbers
Subtract equal inputs and square

Decode the star rule

The two inputs are equal

Test with real numbers

Subtract equal inputs and square

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