Problem 19 · 2023 Math Kangaroo
Medium
Geometry & Measurement
careful-counting
Two rays starting at S form a right angle. More rays starting at S are drawn inside the right angle so that each of the angles 10°, 20°, 30°, 40°, 50°, 60°, 70° and 80° is enclosed by two of the rays. What is the minimum number of rays that have to be drawn inside?
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Answer: B — 3
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Hint 1 of 2
The two arms of the right angle are already drawn; you only add rays inside.
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Hint 2 of 2
Pick a few interior rays so that every listed angle is the gap between two of all the drawn rays.
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Approach: choose interior rays so each required angle is spanned
- The arms at 0° and 90° already exist; the angle between any two rays is the difference of their directions.
- Drawing interior rays at 10°, 40° and 70° gives ray directions 0, 10, 40, 70, 90.
- Their pairwise differences are 10, 20, 30, 40, 50, 60, 70 and 80 — every required angle appears.
- Two interior rays give only a few differences and cannot reach all eight, so the minimum is 3 (B).
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