2008 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.
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 126 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | D | Clocks, Calendars & Time | One third of the job took 2 hours 40 minutes, so the whole job takes three times as long: 8 hours after the 8:30 AM start. | |
| 2 | A | Ratios, Percents & Averages | Only ratios are given, so give the square a side of 1. Then the rectangle is 2 wide and 4 long, and the square fills 1 out of 8. | |
| 3 | A | Divisibility & Factors | The divisors of 6 other than 6 are 1, 2 and 3, which add up to 6. So each step turns 6 back into 6. | |
| 4 | C | Ratios, Percents & Averages | Two thirds of 10 bananas is 20/3 bananas, and that equals 8 oranges. So one banana is worth 6/5 of an orange, and 5/2 bananas are worth 3 oranges. | |
| 5 | B | Algebraic Manipulation | The top of each fraction equals the bottom of the next one, so almost everything cancels, leaving the last top over the first bottom: 2008 over 4. | |
| 6 | D | Ratios, Percents & Averages | Average speed is total distance over total time. With 60 km legs, the times are 20, 3 and 6 hours, so the average is 180 over 29, about 6. | |
| 7 | E | Exponents, Logarithms & Radicals | Top and bottom have the same shape, but every power of 3 on top is 9 times the matching one below. So the whole fraction equals 9. | |
| 8 | A | Ratios, Percents & Averages | Store B takes off 10% more than store A, but A gives 90 dollars back. The 90 beats that 10% by 15, so 10% of the price is 75. | |
| 9 | B | Divisibility & Factors | Over a common bottom of 6 the expression becomes x over 2. So x must be even, and x = 2 shows it need not be a multiple of 3. | |
| 10 | E | Quadrilaterals & Polygon Areas | Joining the midpoints of a square cuts off four corner triangles worth half its area. So each new square is half the last: 16, 8, 4. | |
| 11 | D | Linear Equations & Word Problems | The trip to shore takes 15 minutes, so the boat may gain at most 30 ÷ 15 = 2 gallons each minute. LeRoy must bail the other 8. | |
| 12 | C | Ratios, Percents & Averages | 25% more red than blue means red is 1.25 times blue, so blue is red divided by 1.25, which is 0.8r. Green is 1.6r, and the total is 3.4r. | |
| 13 | D | Linear Equations & Word Problems | Together they paint 1/5 + 1/7 of the room each hour, but they only paint for t minus 1 hours, because one hour of the total is lunch. | |
| 14 | D | Triangles: Area & Pythagorean | A 4 to 3 screen with a 27-inch diagonal is a 3-4-5 triangle scaled up: 21.6 wide and 16.2 tall. The movie is 10.8 tall, leaving 5.4 for two strips. | |
| 15 | D | Linear Equations & Word Problems | Expanding Han's distance shows Ian's speed plus 5 times Ian's time is 65. Jan's extra distance is twice that amount plus 20, which is 150. | |
| 16 | B | Circles | The small center lies on the line halving the 60 degree angle. A 30-60-90 triangle puts it 2r from O, and 2r + r = R, so r = R/3. | |
| 17 | B | Circles | The band around the triangle is three 6 by 3 rectangles, one along each side, plus three rounded corner pieces that fit together into one full circle of radius 3. | |
| 18 | B | Triangles: Area & Pythagorean | Square the sum of the legs: it equals the hypotenuse squared plus 4 times the area. Since that sum is 32 minus the hypotenuse, the squares cancel. | |
| 19 | C | Transformations & Symmetry | Each turn moves P along a quarter circle. Its radius is the diagonal PR, root 40, for the first turn and the side PS = 6 for the second. | |
| 20 | D | Similar & Congruent Triangles | The diagonals cut each other in the ratio 9 to 12, or 3 to 4. Triangles sharing a height have areas in that ratio: 18, 24, 24, 32. | |
| 21 | A | Solid Geometry | All four sides of ABCD are equal, so it is a rhombus. Its diagonals are the cube diagonal, root 3, and a face diagonal length, root 2. | |
| 22 | D | Basic Probability | Only three coin flips matter, so list all eight paths and follow the actual numbers. Five of the eight fourth terms are whole numbers. | |
| 23 | B | Basic Counting | Each letter is in both subsets, only the first, or only the second. Choose the 2 shared letters, split the other 3, then halve. | |
| 24 | D | Modular Arithmetic | k and its square end in 0. Powers of 2 end in 2, 4, 8, 6 repeating. Since k is a multiple of 4, 2^k ends in 6. | |
| 25 | C | Circles | The touching inner corners form a regular hexagon with side x, so the inner edge is known from the center; one right triangle to the table edge gives x. |
2008 AMC 10A. 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.