Problem 1 · 2002 AMC 8
Easy
Geometry & Measurement
spatial-reasoningcareful-counting
A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures?
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Answer: D — 5.
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Hint 1 of 2
Don't try to draw the whole messy picture — every crossing belongs to exactly one *pair* of figures, so count the pairs.
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Hint 2 of 2
Ask the maximum for each shape-pair: line-with-circle, and line-with-line. Add the maxes — that's the most you can ever reach.
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Approach: max crossings = sum over every pair of figures
- Here's the key move: a crossing always happens between *two* figures, so total it pair by pair instead of staring at the tangle. The pairs are line1-circle, line2-circle, and line1-line2.
- A straight line meets a circle in at most 2 points, so the two line-circle pairs give up to 2 + 2 = 4.
- Two lines meet in at most 1 point, adding 1 more. Total: 4 + 1 = 5.
- *You'll see this again:* the most intersections among any set of figures is just the sum of each pair's maximum — count pairs, never the whole picture at once.
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