2002 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 | D | Exponents, Logarithms & Radicals | Take the smallest power of 10 out of the top and out of the bottom. The giant powers cancel, leaving 101 divided by 20, which is about 5. | |
| 2 | C | Functions | Put a = 2, b = 12, c = 9 into the rule in that order, simplify each fraction, then add them using sixths. | |
| 3 | B | Exponents, Logarithms & Radicals | There are only five ways to group four 2s. Because 2 to the 4th and 4 squared are both 16, the five groupings give just two different values. | |
| 4 | E | Diophantine Equations | The question needs just one n for each m. Try n = 1: the rule becomes m is at most m + 1, which is true for every m. | |
| 5 | C | Circles | Along a line through the centers, the big radius is 1 + 2 = 3. Its area, 9 pi, minus seven small circles of area pi leaves 2 pi. | |
| 6 | A | Linear Equations & Word Problems | Undo Cindy's wrong steps backwards to find the starting number, 138. Then do the steps the teacher actually asked for. | |
| 7 | A | Circles | Circle A needs a bigger angle for the same arc, so it is smaller. The radii are in ratio 2 to 3, and areas square that: 4 to 9. | |
| 8 | A | Quadrilaterals & Polygon Areas | Cut the flag into 16 equal small squares. The 4 middle ones are red, and each of the 12 border squares is exactly half white and half blue. | |
| 9 | B | Systems of Equations | Divide both equations by 1001, then add them. The A terms combine to leave exactly A + B + C = 9, so the average is 3. | |
| 10 | A | Quadratics | Both parts contain the same factor, 2x + 3. Pull it out instead of multiplying everything out, and the two roots can be read off directly. | |
| 11 | B | Sets, Estimation & Miscellaneous | Give each big file (0.8 or 0.7) 3 points and each small file 2 points. No disk can hold more than 6 points, but the files total 75 points. | |
| 12 | B | Linear Equations & Word Problems | The two trips differ by 6 minutes. That tells you the distance, 12 miles; then the on-time trip takes 15 minutes, so the speed is 48. | |
| 13 | B | Triangles: Area & Pythagorean | 15-20-25 is 3-4-5 made five times bigger, so it is a right triangle with area 150. The shortest height goes to the longest side: 300 divided by 25 is 12. | |
| 14 | B | Primes | The two roots add to 63, an odd number. Two odd primes would add to an even number, so one root must be 2 and the other 61. | |
| 15 | E | Primes | A two-digit prime cannot end in an even digit or in 5, so 1, 3, 7, 9 must be the last digits and 2, 4, 5, 6 the first digits. | |
| 16 | B | Systems of Equations | Call the shared value x. Then a, b, c, d are x - 1, x - 2, x - 3, x - 4; adding them gives an equation. | |
| 17 | D | Ratios, Percents & Averages | Keep track of coffee and cream separately. Stirred liquid keeps the same mix, so the 3 ounces poured back are 1 ounce of coffee and 2 of cream. | |
| 18 | D | Solid Geometry | Corner dice show 3 faces, edge dice 2, face-center dice 1. Faces 1, 2 and 3 meet at one corner of a die, so each die shows its smallest numbers. | |
| 19 | E | Circles | With the rope free, Spot sweeps 240 degrees of a circle of radius 2. At each neighboring corner the rope bends, and the last yard sweeps another 60 degrees. | |
| 20 | D | Similar & Congruent Triangles | HC and JE are both parallel to AG, so each cuts off a small copy of a bigger triangle: HC is one third of AG, JE one fifth. | |
| 21 | D | Statistics & Data | The mean makes the total 64, and the range makes the smallest number equal the largest minus 8. Push the largest up until the other conditions break. | |
| 22 | C | Number Properties | Starting from a square number of tiles, two rounds always bring you to the next smaller square: 100, 90, 81, then 72, 64, and so on. | |
| 23 | D | Triangles: Area & Pythagorean | The line from E down to the middle of BC has length 8, so AE is the long side of a right triangle with legs 8 and AB + 6. | |
| 24 | A | Basic Probability | Tina has only 10 possible pairs. List each pair's sum; Sergio wins with every number above that sum, which is 10 minus the sum. | |
| 25 | C | Quadrilaterals & Polygon Areas | Slide leg DA over to start at C. It cuts off a 5-12-13 right triangle (13 = 52 - 39), and that triangle's height is the trapezoid's height. |
2002 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.