Problem 28 · 2022 Math Kangaroo
Stretch
Algebra & Patterns
estimate-and-pick
Let N be a positive integer. How many integers are between \(\sqrt{N^2+N+1}\) and \(\sqrt{9N^2+N+1}\)?
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Answer: C — \(2N\)
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Hint 1 of 2
Bound each square root between consecutive integers.
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Hint 2 of 2
√(N^2+N+1) is just above N; √(9N^2+N+1) is just below 3N+1.
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Approach: trap each root between integers
- N^2 < N^2+N+1 < (N+1)^2, so the first root lies between N and N+1; smallest integer above is N+1.
- (3N)^2 < 9N^2+N+1 < (3N+1)^2, so the second root lies between 3N and 3N+1; largest integer below is 3N.
- Integers from N+1 to 3N number 3N − (N+1) + 1 = 2N.
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