Problem 2 · 2022 AMC 8
Easy
Algebra & Patterns
custom-operationsubstitution
Consider these two operations:
a ◆ b = a2 − b2
a ★ b = (a − b)2
What is the value of (5 ◆ 3) ★ 6?
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Answer: D — 100.
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Hint 1 of 2
A strange symbol like ◆ or ★ isn't new math — it's just a made-up recipe. Replace each symbol with its rule and it becomes ordinary arithmetic.
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Hint 2 of 2
Same rule as always: do the parentheses first. Compute the ◆ inside, then drop that number into the ★ recipe.
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Approach: a made-up symbol is just a recipe — replace it with its rule, parentheses first
- Don't be thrown by the unusual symbols: ◆ and ★ are just nicknames for recipes. Wherever you see one, swap in its definition. Same as always, the parentheses go first.
- Inside first: 5 ◆ 3 = 52 − 32 = 25 − 9 = 16.
- Now feed 16 into the ★ recipe: 16 ★ 6 = (16 − 6)2 = 102 = 100.
- Shortcut to notice: 5 ◆ 3 = 52 − 32 is a difference of squares = (5+3)(5−3) = 8 × 2 = 16, so you never even need to square 5 and 3.
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