Problem 15 · 2023 Math Kangaroo
Stretch
Algebra & Patterns
caseworksum-constraint
How many pairs of integers \((m, n)\) fulfil the inequality \(|2m - 2023| + |2n - m| \le 1\)?
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Answer: B — 1
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Hint 1 of 2
The quantity |2m−2023| is always at least 1, since 2m is even and 2023 is odd.
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Hint 2 of 2
So that term must equal exactly 1 and the other term must be 0.
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Approach: use parity to force each absolute value
- 2m−2023 is odd, so |2m−2023| ≥ 1; to satisfy the inequality it must equal 1 and |2n−m| = 0.
- Then m = 1011 or 1012, and m = 2n forces m even, so m = 1012, n = 506.
- That is the only pair, so the answer is 1.
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