2013 AMC 10A

All 25 problems, 75 minutes, scored 6 / 1.5 / 0. The answer key and the idea behind each problem are on this page too, folded away until you ask for them.

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The problems

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This gives away all 25 answers and the idea behind each one. Sit the paper first if you mean to.

#	Answer	Topic	Key insight
1	C	Linear Equations & Word Problems	The fare is a fixed 1.50 plus 0.25 for each of the 5 miles, so add 1.50 and 5 times 0.25.
2	B	Fractions & Decimals	The number of fills is the total amount divided by the cup size: (5/2) divided by (1/4) equals 10.
3	E	Triangles: Area & Pythagorean	Triangle ABE is right-angled at B with legs AB = 10 and BE, so its area is 5 times BE.
4	C	Linear Equations & Word Problems	A game won by scoring twice the opponent's runs needs an even score, so the wins are the even games and the one-run losses are the odd games.
5	B	Linear Equations & Word Problems	Everyone's fair share is the total 405 divided by 3; each payment to Sammy is the gap between 135 and what that person already paid.
6	D	Logic Puzzles	With the 5-year-old home, the movie pair is 3 and 13 or 7 and 9; only 3 and 13 leaves two under-10 brothers for baseball, so Joey is 11.
7	C	Basic Counting	English is forced, so choose 3 of the other 5 courses and throw away the single choice with no math course.
8	C	Exponents, Logarithms & Radicals	Factor 2^2012 out of the numerator and denominator; what remains is (4+1)/(4-1).
9	B	Ratios, Percents & Averages	A three-point attempt yields 0.2 times 3 = 0.6 points and a two-point attempt 0.3 times 2 = 0.6, so every attempt is worth 0.6 regardless of type.
10	E	Ratios, Percents & Averages	Take 100 flowers: 60 pink of which 40 are carnations, 40 red of which 30 are carnations, so 70 carnations.
11	A	Basic Counting	C(n,2) = 10 forces n = 5 council members, and C(5,3) = C(5,2) = 10 by the symmetry of binomial coefficients.
12	C	Similar & Congruent Triangles	Parallel sides copy the base angles, so triangles DBE and FEC are isosceles: BD = DE and EF = FC, and the perimeter collapses to AB + AC.
13	B	Bases & Digits	Numbers are aba with a not 0 or 5; every b works for a at most 4, but a from 6 to 9 needs b below 20-2a.
14	D	Solid Geometry	Each of the 12 original edges keeps its middle third, and each of the 8 corner notches contributes 9 new edges: 12 + 72 = 84.
15	D	Triangles: Area & Pythagorean	Every altitude equals 2K divided by its side, so the condition becomes 1/c = (1/10 + 1/15)/2 and the area cancels.
16	E	Coordinate Geometry	The reflected vertex (10,5) lies on line BC extended, so the union is the big triangle with base from (6,5) to (10,5) minus the small triangle above the crossing point.
17	B	Inclusion-Exclusion	Two specific friends coincide on multiples of their lcm (12, 15 or 20); subtract the multiples of 60, where all three come, from each count.
18	B	Coordinate Geometry	The piece on the D side is triangle APD with base AD = 4, so area 15/4 fixes the height of P as 15/8; then P is on line CD.
19	C	Bases & Digits	The last base-b digit of 2013 is its remainder mod b, so b must divide 2010 = 235*67 (16 divisors) with b > 3.
20	C	Transformations & Symmetry	Vertices sweep 45-degree arcs of the circumcircle, and between those arcs the boundary is just the sides of the starting and ending squares.
21	D	Divisibility & Factors	After pirate k the chest holds (12-k)/12 of what it held, so the last pirate gets N*11!/12^11; the minimal N cancels all of 12^11 not covered by 11!.
22	B	Solid Geometry	The big sphere has radius 3; the eighth center sits on the axis at height 3 - r, and tangency to a small sphere gives (3-r)^2 + 4 = (r+1)^2.
23	D	Circles	Power of point C gives CX * CB = 97^2 - 86^2 = 2013 = 31161; the triangle inequality forces 11 < BC < 183, leaving only 33 * 61.
24	E	Games & Processes	Each round is one of 6 pairings, which split into two families of 3; every game lies in one pairing from each family, so usage counts are forced.
25	A	Basic Counting	Every 4 vertices give one crossing pair, C(8,4) = 70, but the center is counted 6 times and eight triple points are counted 3 times each.

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic tags, difficulty ratings and key insights on this page are original to this site. No problem statements are reproduced here — each links to its own page.