Problem 13 · 2022 AMC 8
Medium
Algebra & Patterns
substitutionsum-constraint
How many positive integers can fill the blank in the sentence below?
"One positive integer is ___ more than twice another, and the sum of the two numbers is 28."
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Answer: D — 9 values.
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Hint 1 of 2
The blank isn't free to be anything — once you choose the smaller number, the sum of 28 forces the blank. So really you're counting how many smaller numbers are allowed.
Still stuck? Show hint 2 →
Hint 2 of 2
Smaller = a, larger = 2a + (blank). Their sum 28 gives blank = 28 − 3a. Now ask: which a keep both the blank and the numbers positive?
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Approach: the blank is determined by the smaller number, so count valid smaller numbers
- Insight: don't search for blanks — each choice of the smaller number forces the blank. Let smaller = a; then larger = 2a + (blank), and the sum is 3a + (blank) = 28, so blank = 28 − 3a.
- For the blank to be a positive integer, 28 − 3a ≥ 1, i.e. a ≤ 9; and a ≥ 1. Each such a gives a different blank, so the count of blanks equals the count of a.
- a ∈ {1, 2, …, 9} ⇒ 9 values for the blank.
- Sanity check: a = 1 gives blank 25 (numbers 1 and 27); a = 9 gives blank 1 (numbers 9 and 19) — both endpoints valid. ✓
Mark:
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