2011 AMC 10B

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)Print the paper PDF 165 KB 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.

The same key as a PDF 127 KB — a separate file from the paper, so one can be handed over without the other.

#	Answer	Topic	Key insight
1	C	Fractions & Decimals	Add up the tops and bottoms first. The problem becomes 12/9 minus 9/12, which is 4/3 minus 3/4.
2	E	Ratios, Percents & Averages	Her average now is 77, so she wants 80. Six tests averaging 80 must add up to 480, and she already has 385.
3	A	Absolute Value & Inequalities	Each side could be up to half an inch shorter than reported. The smallest area uses the shortest possible width and the shortest possible length together.
4	C	Linear Equations & Word Problems	Each person should pay half of the total. LeRoy gives Bernardo the difference between that half and what he already paid.
5	E	Divisibility & Factors	Break 161 into factors: it is 7 times 23. The reversed number has two digits, so it was 23. That makes the real number 32 and the other number 7.
6	A	Linear Equations & Word Problems	Undo the story backwards from the end. Add back what he gave away, then remember what is left after eating a third is two thirds of what he had.
7	B	Angles & Polygons	The two angles add to 108 degrees, so they are 39 and 69. The third angle is 180 minus 108, which is 72, and it is the biggest.
8	B	Logic Puzzles	Hot and sunny always means crowded. So if the beach was not crowded, the day was not both hot and sunny: at least one of the two was missing.
9	D	Similar & Congruent Triangles	Triangle EBD is a smaller copy of ABC. One third of the area means sides shrink by the square root of 3, and BD matches BC = 4.
10	B	Sequences & Series	The ten smaller numbers add up to 1,111,111,111. Nine times that is 9,999,999,999, just 1 short of the largest number, so the ratio is almost exactly 9.
11	D	Sets, Estimation & Miscellaneous	If every month had at most 4 birthdays, the room would hold at most 48 people. With 52, some month has 5, and 6 can fail.
12	A	Circles	The straight parts are equal on both edges. The two round ends form a circle whose radius grows by 6, adding exactly 12 pi meters.
13	D	Geometric Probability	The product is positive when both numbers are negative or both are positive. A number is negative 2/3 of the time and positive 1/3 of the time.
14	C	Algebraic Manipulation	The diagonal gives l squared plus w squared = 625, and the area gives lw = 168. So (l + w) squared = 625 + 336 = 961.
15	E	Functions	Replace each @ by 'add, then divide by 2' and simplify. Law I gets an extra half of x; laws II and III come out equal.
16	A	Geometric Probability	Let the octagon's side be the square root of 2. The corner triangles then have legs 1, and the octagon is a square minus four corners.
17	C	Circles	Angle A is 90 degrees because EB is a diameter, so angle ABE is 50. Parallel lines copy that 50 to angle BED, and angle BCD is 180 minus 50.
18	E	Triangles: Area & Pythagorean	Parallel sides make angle MDC equal angle AMD, so triangle MDC has two equal sides: CM = CD = 6. With CB = 3, angle CMB is 30.
19	A	Absolute Value & Inequalities	Let u stand for \|x\|. The equation becomes u squared minus 5u minus 24 = 0, so \|x\| = 8. The roots are 8 and -8.
20	C	Quadrilaterals & Polygon Areas	The short diagonal BD splits the rhombus into two equilateral triangles of side 2. In each, the part closer to B than to the other corners is exactly one third.
21	B	Number Properties	The biggest difference, 9, is w minus z. It is made of three gaps between neighbors, which must be 1, 3 and 5, with 1 in the middle.
22	A	Solid Geometry	The pyramid's height is half the square root of 2. The level slice at the cube's height is a square, and it is exactly the cube's top face.
23	D	Modular Arithmetic	Only the last three digits matter, and they match those of 11 to the 2011th power. For 11 to the n, they come from 1 + 10n + 50n(n-1).
24	B	Coordinate Geometry	The line hits a lattice point at x exactly when the slope times x is whole. So a is the smallest fraction above 1/2 with bottom at most 100.
25	D	Circles	Each new side is half the perimeter minus an old side. Three sides in a row stay three sides in a row, and the middle side halves each time.

2011 AMC 10B. Problems © Mathematical Association of America, American Mathematics Competitions (maa.org/amc), licensed CC BY-NC-SA 4.0. Typeset and annotated by AMC 10 Step by Step (amcstep.com) and shared under the same licence. Not endorsed by the MAA.