2011 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.

Take it timed (75 min)Blank answer sheet Enter answers from paper

The problems

Show the answer key, topics and key insights

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	D	Linear Equations & Word Problems	Only the half hour beyond 30 hours is billed, and it must be converted to 30 minutes at 10 cents each before adding the text charge.
2	E	Number Properties	Fourteen small bottles hold only 490 milliliters, short of 500, so a fifteenth bottle is required; the answer is the ceiling of 500/35.
3	D	Functions	Evaluate the innermost brackets first: {1 1 0} = 2/3 and [0 1] = 1/2, then average those two with 0.
4	A	Sequences & Series	The two sums share every term from 12 to 100, so Y minus X is just the leftover 102 minus the leftover 10.
5	C	Ratios, Percents & Averages	The grades are in the ratio 4:2:1, so take 4, 2 and 1 students and compute a weighted average: (48 + 30 + 10)/7.
6	C	Sets, Estimation & Miscellaneous	The union is smallest when the smaller set sits entirely inside the larger one, so it cannot have fewer than 20 elements and 20 is attainable.
7	B	Absolute Value & Inequalities	An absolute value is never negative, so \|-3x\| + 5 is at least 5 and can never equal 0; every other equation has an explicit solution.
8	C	Ratios, Percents & Averages	Removing the swans leaves 75% of the birds, and the geese are 30% of the original total, so the fraction is 30/75 = 40%.
9	A	Coordinate Geometry	The lines sit on opposite sides of the axes, so the rectangle is (c + d) by (a + b); expanding gives four positive terms.
10	B	Divisibility & Factors	1771 = 7 * 11 * 23 and the number of buyers must be between 16 and 30, so 23 students bought pencils, leaving 7 pencils at 11 cents.
11	B	Triangles: Area & Pythagorean	With side 8, the four corner triangles are congruent right triangles with legs 7 and 1, so the inner square has area 7^2 + 1^2 = 50 out of 64.
12	A	Linear Equations & Word Problems	Let y be the number of two-pointers; then three-pointers also contribute 2y points and free throws contribute y + 1, so 5y + 1 = 61.
13	A	Basic Counting	Hundreds digit is 2 or 5, units digit is 2 or 8, and hundreds digit 2 leaves only 8 for the units place.
14	B	Basic Probability	Area below circumference means pi d^2/4 < pi d, i.e. d < 4, so the dice sum must be 2 or 3: three of the 36 outcomes.
15	C	Linear Equations & Word Problems	Average mpg is total miles over total gallons, and the gallons come only from the last x - 40 miles: x / (0.02(x - 40)) = 55.
16	B	Exponents, Logarithms & Radicals	Square the whole sum: the cross term is 2 sqrt((9 - 6 sqrt 2)(9 + 6 sqrt 2)) = 2 sqrt(9) = 6, so the square is 24.
17	C	Sequences & Series	Subtracting overlapping triple sums gives A = D = G and B = E = H, so A + H = A + B = 30 - 5.
18	C	Circles	Center C is sqrt(2) from A and B, so circle C overlaps each of them in a 90-degree lens of area pi/2 - 1.
19	E	Number Properties	The 2001 condition gives b^2 - a^2 = 141 = 1141 = 347, so a is 70 or 22; only a = 22 makes a^2 + 300 a perfect square.
20	D	Geometric Probability	A chord of length r spans 60 degrees; fix the first point, and the chords cross exactly when the second point lands within 60 degrees on either side.
21	D	Conditional Probability & States	The pairs weigh the same only when both are all genuine or each contains exactly one counterfeit, so compare just those two counts.
22	C	Arrangements with Restrictions	The five diagonals form one closed loop A-C-E-B-D-A, so count proper colorings of a 5-cycle: 6*5^4 minus the proper colorings of a 4-cycle.
23	C	Games & Processes	The unsaid numbers always form an arithmetic progression; each skip sends (first term, step) to (a + d, 3d), producing 2, 5, 14, 41, 122, 365.
24	D	Solid Geometry	Each tetrahedron's edges are face diagonals meeting at face centers, so one tetrahedron's faces slice half-scale corners off the other, leaving 1/3 - 4/24 = 1/6.
25	C	Triangles: Area & Pythagorean	Rays must hit the four corners, forcing both coordinates of an n-ray partitional point to be multiples of 2/n; count the 49-by-49 grid minus the 9-by-9 overlap.

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.