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2002 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2002 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 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 — answer key. 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.