2008 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.

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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 125 KB — a separate file from the paper, so one can be handed over without the other.

#	Answer	Topic	Key insight
1	E	Basic Counting	Each time a 2-point basket becomes a 3-point basket, the total goes up by exactly 1. Six choices for the number of 3-pointers give six totals.
2	B	Sets, Estimation & Miscellaneous	Write out the new block after reversing the two rows, then add each diagonal. The sums are 50 and 54.
3	D	Exponents, Logarithms & Radicals	Turn every root into a fraction exponent. x times the square root of x is x to the 3/2, and a cube root divides that exponent by 3.
4	C	Linear Equations & Word Problems	To make one salary as large as possible, pay everyone else the smallest amount allowed. The star gets whatever is left of the 700,000.
5	A	Functions	(x - y) squared and (y - x) squared are the same number. The operation squares the difference of its two inputs, and equal inputs have difference 0.
6	C	Ratios, Percents & Averages	Measure both points from D. BD is one fifth of AD and CD is one tenth of AD, so BC is the difference, one tenth of AD.
7	C	Similar & Congruent Triangles	Count the small triangles row by row: 1, 3, 5, and so on up to 19. The first ten odd numbers add to 10 times 10, which is 100.
8	C	Diophantine Equations	Carnations cost an even amount and 50 is even, so the roses must cost an even amount too. The number of roses is 0, 2, 4, up to 16.
9	A	Quadratics	Divide by a to get x squared minus 2x plus b over a. The two solutions add to 2, so their average is 1, no matter what b is.
10	A	Circles	Draw a line from the center through the chord's midpoint. The chord is 4 from the center and C is 1 beyond it, so AC has legs 3 and 1.
11	B	Sequences & Series	The missing term between the two known ones is the fourth term. Call it a, build the fifth and sixth terms from it, and set the sixth equal to 128.
12	A	Sets, Estimation & Miscellaneous	Each flip means 100,000 steps. Forty-four flips plus the final 50,000 give 4,450,000 steps, and dividing by 1800 gives about 2470 miles.
13	B	Sequences & Series	If the first n terms average n, they add up to n times n. Each term is the jump from one square to the next: the odd numbers.
14	B	Coordinate Geometry	A sits straight above B at height 5 divided by the square root of 3. A quarter turn counterclockwise sends (x, y) to (-y, x).
15	A	Diophantine Equations	The Pythagorean theorem simplifies to a squared equals 2b plus 1. So a is odd, and b stays below 100 only up to a = 13.
16	A	Basic Probability	Once at least one die is rolled, the sum is odd half the time. Only two tails is different: no dice, and 0 is even.
17	B	Basic Probability	One fixed order, like yes-no-no, has chance 0.7 times 0.3 times 0.3. The single yes can come on any of 3 occasions, so multiply by 3.
18	B	Linear Equations & Word Problems	Measure speed in bricks per hour. Brenda lays one ninth of the chimney each hour and Brandon one tenth; together, minus 10 bricks, they must lay one fifth.
19	E	Solid Geometry	Look at the tank's round end. The water is a one-third slice of the circle minus a triangle. Multiply that area by the length 9.
20	B	Basic Probability	Each die has six faces, so there are 36 equally likely outcomes. Count the face pairs for sums 5, 7 and 9, counting a repeated number once for each face.
21	C	Arrangements with Restrictions	Seat the men first. Each wife then has exactly two allowed chairs, and one wife's choice forces all the others, so the wives can sit in just 2 ways.
22	C	Basic Probability	Place the three red beads first. They cannot touch, so they fit only four ways in six spots. Then check where the white and blue beads can go.
23	B	Diophantine Equations	The painted part is 2 feet shorter in each direction and is half the floor. The equation rearranges to (a minus 4) times (b minus 4) equals 8.
24	C	Angles & Polygons	Cut angles B and C in half with lines meeting at E. This makes three matching triangles with 60-degree angles at E, so E lands on side AD.
25	B	Linear Equations & Word Problems	Track the truck's lead over Michael. Each 50-second cycle it gains 100 feet while driving and loses 150 while stopped, so the lead shrinks by 50.

2008 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.