AMC 8 · 2002 · #1

Grade 7 geometry-2dcounting
Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation
spatial-visualizationsystematic-enumerationcombinations-basic identify-subproblemssystematic-enumeration ↑ Prerequisites: systematic-enumeration
📏 Short solution 💡 2 insights
Problem
A circle and two distinct straight lines are drawn on the same sheet of paper. Counting every point where any two of these three figures cross, what is the largest possible total number of intersection points?

Pick an answer.

(A)
2
(B)
3
(C)
4
(D)
5
(E)
6

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Break into Subproblems

Three figures make three pairs: line-line, line-circle (twice). Tool #7 (Break into Subproblems) lets us handle one pair at a time, find the maximum for each, and add. Tool #15 (Visualize) is needed at the end to draw a single picture where every pair hits its maximum at the same time, so the five points are all different and the total is actually reachable.

1STEP 1

Pair 1, line vs line: two distinct lines cross at most once, so cross them for the max of 1 point.

line ∩ line ≤ 1 point
2STEP 2

Pair 2, circle vs first line: a secant through the inside cuts the circle at the max of 2 points.

circle ∩ line₁ ≤ 2 points
3STEP 3

Pair 3, circle vs second line: independent of the first, it is also a secant for another 2 points.

circle ∩ line₂ ≤ 2 points
4STEP 4

Add the three pair-maximums, 1 + 2 + 2, to bound the total at 5.

1 + 2 + 2 = 5
5STEP 5

Sketch a circle with two secants crossing inside it: 1 inner crossing + 4 on the circle = 5 distinct points, so the bound is reached.

achievable total = 5 → (D)
Answer
5
Counting cross-checks: with 3 figures there are C(3, 2) = 3 pairs, and the per-pair maxes are 1, 2, 2, which sum to 5 — matching choice (D). Pushing further is impossible: a third intersection on either line-circle pair would force a line to meet a circle in 3 points (no such line exists), and a second line-line intersection would force the two lines to coincide (not distinct). So 5 is both an upper bound and reachable.
💡Key takeaway

Three figures make three pairs. Find each pair's biggest intersection count, add them up, and then make sure one picture can hit all the maximums at once — that final check is what turns 5 from a guess into the real answer.

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