2004 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 123 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | A | Linear Equations & Word Problems | Count the people first. You plus five friends is six people, so each person raises 1500 divided by 6. | |
| 2 | B | Functions | Work from the inside out. The three inner values are -1, 1 and -3; then use the same rule once more on those three numbers. | |
| 3 | E | Ratios, Percents & Averages | The answer must be in cents, so change 20 dollars into 2000 cents first. Then 1.45 percent of 2000 is 29. | |
| 4 | D | Absolute Value & Inequalities | The distance from x to 1 equals the distance from x to 2, so x is exactly halfway between 1 and 2, which is 3/2. | |
| 5 | C | Basic Probability | There are 84 ways to pick 3 of the 9 points, and only 8 of them lie on a line: the 3 rows, the 3 columns and the 2 diagonals. | |
| 6 | E | Linear Equations & Word Problems | The 24 granddaughters come in groups of 6, so only 4 daughters are mothers. Everyone else, 30 - 4 = 26 women, has no daughters. | |
| 7 | C | Sequences & Series | Each orange sits in a gap between four below, so each layer shrinks by one both ways: 5 by 8, 4 by 7, down to 1 by 4. | |
| 8 | B | Games & Processes | Every three rounds, each player ends up with exactly one fewer token. So after 36 rounds they hold 3, 2, 1, and round 37 leaves A with nothing. | |
| 9 | B | Triangles: Area & Pythagorean | Both big right triangles contain the same piece, triangle ABD. Take it away from each and the difference stays the same: 16 - 12 = 4. | |
| 10 | D | Basic Probability | Go through each possible number of heads, 0 to 3. Multiply the ways coin A and coin B can each get that many, then add: 35 out of 128. | |
| 11 | C | Ratios, Percents & Averages | Making the jar 5/4 as wide makes the base 25/16 as big, so the height must shrink to 16/25 of what it was. That is a 36 percent drop. | |
| 12 | C | Basic Counting | For each of the 8 condiments you just say yes or no, giving 2 to the 8th = 256 choices. Multiply by 3 patty choices. | |
| 13 | D | Basic Counting | Count dance pairs two ways. The men make 12 times 3 = 36 pairs; each woman is in 2 pairs, so there are 18 women. | |
| 14 | A | Ratios, Percents & Averages | Adding one 25-cent coin raises the average by exactly 1 cent only if there were 4 coins, worth 80 cents in all. That forces three quarters and a nickel. | |
| 15 | D | Absolute Value & Inequalities | Split the fraction into 1 + y/x. Since y/x is always negative, make it as close to 0 as possible: y = 2 and x = -4 give 1/2. | |
| 16 | D | Basic Counting | Count by size. The numbers of squares that cover the center are 1, 4, 9, 4, 1 for sizes 1 through 5, which adds to 19. | |
| 17 | C | Linear Equations & Word Problems | Together they run half a lap before the first meeting and a full lap between meetings, so Brenda runs 200 more; 200 + 150 is one lap. | |
| 18 | A | Sequences & Series | The new numbers are 9, 11 + d and 29 + 2d. Making them geometric gives d = 10 or d = -14; the second gives third term 1. | |
| 19 | C | Solid Geometry | Unroll the silo into a flat rectangle. The stripe becomes slanted strips 3 feet wide climbing 80 feet in total, so its area is 3 times 80. | |
| 20 | D | Triangles: Area & Pythagorean | With side 1 and AE = CF = t, setting sides BE and EF equal gives (1 - t) squared = 2t, which makes the ratio exactly 2. | |
| 21 | B | Circles | The shaded area works out to 3 pi + 3 theta, and it is 8 parts out of 8 + 13 = 21 parts of the whole 9 pi. | |
| 22 | D | Circles | Sides BC and AD also touch the semicircle, so equal tangent lengths give CE = 2 + x and DE = 2 - x; the Pythagorean theorem finds x. | |
| 23 | D | Circles | Circle D has radius 2. Put B's center at (x, r); touching D and touching A give two distance equations, leading to r = 8/9. | |
| 24 | D | Functions | Climb the powers of 2 using the rule. Reaching 2 to the 100th multiplies factors 2 to the 0, 1, up to 99, so the exponent is 4950. | |
| 25 | B | Solid Geometry | Use the centers. The big center is 3 from each small center, above the middle of an equilateral triangle of side 2, 2 over root 3 from each corner. |
2004 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.