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.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | D | Exponents, Logarithms & Radicals | Pull the smallest power of 10 out of the top and bottom; the huge exponents cancel and only 101/20 remains. | |
| 2 | C | Functions | Substitute a=2, b=12, c=9 into the definition and add the three fractions over the common denominator 6. | |
| 3 | B | Exponents, Logarithms & Radicals | There are only five ways to parenthesize four 2s; since 2^(2^2) = (2^2)^2 = 16, every grouping evaluates to either 2^16 or 256. | |
| 4 | E | Diophantine Equations | Taking n = 1 turns the inequality into m <= m + 1, which is true for every positive integer m. | |
| 5 | C | Circles | A diameter of the big circle passes through three unit circles in a row, so its radius is 3; subtract seven unit-circle areas from 9 pi. | |
| 6 | A | Linear Equations & Word Problems | Undo Cindy's steps in reverse (multiply 43 by 3, add 9) to recover the original number 138, then apply the correct steps. | |
| 7 | A | Circles | Equal arc lengths mean 45 r_A = 30 r_B, so the radii are in ratio 2:3 and the areas in ratio 4:9. | |
| 8 | A | Quadrilaterals & Polygon Areas | Cut the flag into a 4-by-4 grid of equal squares; the 12 border squares are each exactly half white and half blue, so B = W. | |
| 9 | B | Systems of Equations | Add the two equations: the A terms combine to 1001A, so 1001(A+B+C) = 9009 and the average follows without finding A, B, C. | |
| 10 | A | Quadratics | Both terms share the factor (2x+3); factor it out instead of expanding, and the two roots appear immediately. | |
| 11 | B | Sets, Estimation & Miscellaneous | Only 0.7+0.7, 0.8+0.4, 0.7+0.4 and 0.4+0.4+0.4 fit on a disk; the 15 large files force at least 9 disks with little room for the 15 small ones. | |
| 12 | B | Linear Equations & Word Problems | The two trips differ by 6 minutes, so d/40 - d/60 = 1/10 hour gives d = 12 miles and the on-time trip takes 15 minutes. | |
| 13 | B | Triangles: Area & Pythagorean | 15-20-25 is a right triangle, so the area is 150, and the shortest altitude is the one to the longest side: 2*150/25 = 12. | |
| 14 | B | Primes | The roots add to the odd number 63, so one prime root must be the only even prime, 2, forcing the other to be 61. | |
| 15 | E | Primes | Two-digit primes cannot end in 2, 4, 5 or 6, so those are the tens digits and 1, 3, 7, 9 the units digits; the sum is forced. | |
| 16 | B | Systems of Equations | Name the common value x; then a, b, c, d are x-1, x-2, x-3, x-4, and their sum 4x-10 must also equal x-5. | |
| 17 | D | Ratios, Percents & Averages | Track ounces of coffee and cream separately: after the pour-back, cup 1 holds 3 oz coffee and 2 oz cream, so cream is 2/5 of 5 oz. | |
| 18 | D | Solid Geometry | Classify dice by visible faces: 8 corners show 3 mutually adjacent faces (min 1+2+3), 12 edges show 2 (min 1+2), 6 face-centers show 1; total 8*6+12*3+6*1 = 90. | |
| 19 | E | Circles | The reachable region is a 240-degree sector of radius 2 plus, after the rope wraps around each neighboring vertex, two 60-degree sectors of radius 1. | |
| 20 | D | Similar & Congruent Triangles | Both HC and JE are parallel to AG, so triangles DCH and FEJ are scaled copies of DAG and FAG with ratios 1/3 and 1/5; divide. | |
| 21 | D | Statistics & Data | Mean fixes the sum at 64, range ties the minimum to max minus 8, and unique mode forces enough 8s to outnumber the repeated low values. | |
| 22 | C | Number Properties | From n^2 tiles, two operations remove n and then n-1 tiles, landing exactly on (n-1)^2; so going from 10^2 down to 1^2 takes 2*9 = 18 steps. | |
| 23 | D | Triangles: Area & Pythagorean | The altitude from E to BC has length 8, so AE = sqrt((x+6)^2 + 64), and the perimeter condition AE = 26 - x squares to a linear equation. | |
| 24 | A | Basic Probability | Tina has only 10 possible pairs; tabulate each pair's sum s, and Sergio wins in exactly 10 - s ways, so the answer is a short sum divided by 100. | |
| 25 | C | Quadrilaterals & Polygon Areas | Slide the two legs together: the legs 5 and 12 with base 52 - 39 = 13 form a 5-12-13 right triangle, whose altitude 60/13 is the trapezoid's height. |
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.