Problem 15 · 2023 Math Kangaroo
Medium
Geometry & Measurement
Counting & Probability
spatial-reasoningcareful-counting
The side lengths of a square are 1 cm long. How many points in the plane are there that are exactly 1 cm away from two corner points of the square?
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Answer: E — 12
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Hint 1 of 2
A point 1 cm from a given corner lies on a circle of radius 1 centred at that corner.
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Hint 2 of 2
Count, for every pair of corners, where their two unit circles cross — both adjacent pairs and diagonal pairs give crossings.
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Approach: intersect unit circles centred at the corners, over all corner pairs
- A point 1 cm from two corners lies where the two radius-1 circles around those corners meet.
- The 4 adjacent corner pairs are 1 cm apart, so each pair's circles cross in 2 points: 4 × 2 = 8 points.
- The 2 diagonal pairs are √2 cm apart (< 2), so each also crosses in 2 points: 2 × 2 = 4 points.
- Altogether 8 + 4 = 12 points, so the answer is 12 (E).
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