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.
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 | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | E | Basic Counting | With k three-point baskets the total is 10 + k, and k can be any of 0 through 5, so there are six totals. | |
| 2 | B | Sets, Estimation & Miscellaneous | Just rewrite the block after the two reversals and add the diagonals: 1+10+17+22 = 50 versus 4+9+16+25 = 54. | |
| 3 | D | Exponents, Logarithms & Radicals | Convert to fractional exponents: x times sqrt(x) is x^(3/2), and the cube root divides the exponent by 3. | |
| 4 | C | Linear Equations & Word Problems | To maximize one salary, pay the other 20 players the minimum 15,000 each and give the rest of the 700,000 cap to the star. | |
| 5 | A | Functions | Since (x-y)^2 and (y-x)^2 are the same number, the operation is applied to two equal inputs and gives (0)^2 = 0. | |
| 6 | C | Ratios, Percents & Averages | Both conditions are measured from D: BD is 1/5 of AD and CD is 1/10 of AD, so BC = BD - CD = 1/10 of AD. | |
| 7 | C | Similar & Congruent Triangles | The big triangle is a scaled copy of the small one with ratio 10, so its area is 10^2 = 100 times as large. | |
| 8 | C | Diophantine Equations | In 3r + 2c = 50 the term 3r must be even, so r is even; r can be 0, 2, ..., 16, which is 9 values. | |
| 9 | A | Quadratics | By Vieta the roots sum to 2a/a = 2, so their average is 1 regardless of b. | |
| 10 | A | Circles | The line from the center through the arc midpoint bisects chord AB perpendicularly; the center is 4 from the chord, so C is 1 beyond it. | |
| 11 | B | Sequences & Series | Call u_4 = a; then u_5 = 2a + 9 and u_6 = 5a + 18, so 5a + 18 = 128 gives a = 22 and u_5 = 53. | |
| 12 | A | Sets, Estimation & Miscellaneous | Each flip is 100,000 steps, so the year totals 44(100,000) + 50,000 = 4,450,000 steps; divided by 1800 that is about 2470 miles. | |
| 13 | B | Sequences & Series | Mean n over n terms means the partial sums are n^2, so the n-th term is n^2 - (n-1)^2 = 2n - 1. | |
| 14 | B | Coordinate Geometry | A = (5, 5/sqrt3) from the 30-60-90 triangle, and a 90-degree counterclockwise rotation sends (x, y) to (-y, x). | |
| 15 | A | Diophantine Equations | The Pythagorean condition collapses to a^2 = 2b + 1, so a is odd and b = (a^2 - 1)/2 < 100 forces a in {3, 5, ..., 13}. | |
| 16 | A | Basic Probability | With at least one die the sum is odd half the time; no dice (probability 1/4) gives the even sum 0, so the answer is 3/4 times 1/2. | |
| 17 | B | Basic Probability | Exactly one approval in three independent trials: choose which occasion (3 ways) times 0.7 times 0.3 squared, giving 0.189. | |
| 18 | B | Linear Equations & Word Problems | In bricks per hour the rates are N/9 and N/10; together minus 10 they equal N/5, and 1/9 + 1/10 - 1/5 = 1/90 gives N = 900. | |
| 19 | E | Solid Geometry | The water cross-section is a circular segment cut by a chord 2 below the center: a 120-degree sector minus a triangle, then times the length 9. | |
| 20 | B | Basic Probability | The targets 5, 7, 9 are all odd sums except 3 and 11; 18 of 36 outcomes are odd, and 3 and 11 each occur twice, leaving 14. | |
| 21 | C | Arrangements with Restrictions | Seat the men first (2 parities times 5!); each wife then has exactly 2 legal chairs, and that chain of options forces exactly 2 full seatings. | |
| 22 | C | Basic Probability | Place the three reds in non-adjacent slots first (4 ways among 6 positions), then check how the two whites and the blue can fill the gaps without whites touching. | |
| 23 | B | Diophantine Equations | The painted rectangle is (a-2) by (b-2) and equals half the floor; clearing gives ab - 4a - 4b + 8 = 0, which factors as (a-4)(b-4) = 8. | |
| 24 | C | Angles & Polygons | The bisectors of angles B and C meet at E with three congruent triangles and three 60-degree angles at E, so E lies on AD and angle BAD = 85. | |
| 25 | B | Linear Equations & Word Problems | Track the truck's lead over each 50-second cycle: up 100 while driving, down 150 while stopped, so it drifts down 50 per cycle and crosses zero finitely often. |
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.