AMC 10 · 2002 · #18
Grade 7 arithmeticPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Four circles crossing all at once looks tangled, but the crossings never happen between three circles — every crossing point belongs to exactly one pair of circles. Tool #7 (Identify Subproblems) uses that to split the whole count into a sum over pairs: the total is just (points per pair) times (number of pairs). Tool #1 (Draw a Diagram) pins down the first factor — two circles cross at most twice. Tool #2 (Make a Systematic List) pins down the second — carefully list the pairs of four circles so none is missed or double-counted. Tool #14 (Extreme Principle) closes the argument: the total is largest when every pair truly meets twice and no three circles share a point, and we check that this best case is actually drawable.
Two circles cross at most twice
Two distinct circles can miss, touch once, or cross twice — never three times, so one pair gives at most 2 points.
A circle is fixed once you know its center and radius, so two circles just can't wrap around each other enough to cross a third time.
Two circles can cross at most twice, because a circle is fixed once its centre and radius are known.
▸ Why?
Every point of a circle sits one fixed distance from its centre, which leaves no room for a third crossing.
▸ Why?
Each crossing belongs to exactly one pair of circles, so counting pairs counts crossings without repeats.
Every crossing belongs to one pair
A crossing sits on two circles, so it is owned by exactly one pair — the grand total is the pair-by-pair counts added up.
Splitting a messy count into one tidy count per pair turns a tangle into simple arithmetic.
4.OA.A.3Identify SubproblemsList the pairs of circles
Label the circles 1, 2, 3, 4 and list each pair once, smaller first: {1,2},{1,3},{1,4},{2,3},{2,4},{3,4} — 6 pairs.
Listing pairs in order — smaller number first — guarantees you count each pair exactly once.
7.SP.C.8Make A Systematic ListMultiply, then check it is reachable
Six pairs at 2 points each caps the count, and four equal circles centered on a small square reach it: 6 × 2 = 12, choice (D).
The most points appears when every pair is pushed to its limit of two and no crossing is shared away.
3.OA.A.1Extreme PrincipleTwo circles can cross at most twice, so count how many pairs of circles there are and double it.
- Two circles cross at most twice
- Every crossing belongs to one pair
- List the pairs of circles
- Multiply, then check it is reachable