Problem 25 · 2020 AMC 8
Hard
Algebra & Patterns
substitutionsum-constraint
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Answer: A — S2 has side 651.
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Hint 1 of 2
You have three unknown side lengths but only want s2. Don't try to solve for all three — look for two expressions (the width and the height) where the other two unknowns will cancel when you subtract.
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Hint 2 of 2
Width = s1 + s2 + s3. The height threads through R2 and S3; since R2's height is s1 − s2, the height = s1 − s2 + s3. Notice s2 appears with opposite signs in the two.
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Approach: build width and height so subtracting kills s1 and s3
- Across the top, the three squares span the full width: s1 + s2 + s3 = 3322.
- Down the right side, the height is R2's height plus s3. From the figure, R2's height equals s1 − s2, so the height is s1 − s2 + s3 = 2020.
- Subtract the second from the first: the s1 and s3 vanish and the two s2 terms add: 2s2 = 3322 − 2020 = 1302, so s2 = 651.
- Why this transfers: with more unknowns than you care about, don't solve the whole system — combine equations so the unwanted variables cancel. Here width and height were engineered so that subtracting wiped out s1 and s3 in one stroke, leaving only the target.
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