2003 AMC 10B
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 | C | Algebraic Manipulation | Every term on top is 2/3 of the term directly below it, so the common alternating sum cancels and the fraction is 2/3. | |
| 2 | D | Linear Equations & Word Problems | Fourteen days cost 546 dollars, so one green plus one pink pill costs 39, and two prices differing by 1 that sum to 39 are 20 and 19. | |
| 3 | B | Sequences & Series | The first 8 odd numbers sum to 64, so the five even integers sum to 60; their middle term is the average 12, so the smallest is 8. | |
| 4 | A | Quadrilaterals & Polygon Areas | Read the five rectangle areas off the figure (21, 20, 15, 6, 4), then pair the cheapest flower with the largest region and so on down. | |
| 5 | C | Linear Equations & Word Problems | The overlap makes each pass effectively 24 inches = 2 feet wide, so Moe clears 5000 times 2 = 10000 square feet per hour against a 13500 square foot lawn. | |
| 6 | D | Triangles: Area & Pythagorean | A 4:3 rectangle has a 5-unit diagonal in the same units, so the horizontal length is 4/5 of the 27-inch diagonal, about 21.6. | |
| 7 | B | Number Properties | Floor of sqrt(n) equals k for the 2k+1 integers from k^2 to (k+1)^2 - 1, so group the sum as 3 ones, 5 twos, 7 threes, and one 4. | |
| 8 | B | Sequences & Series | Two terms apart in a geometric sequence differ by a factor r^2 = 6/2 = 3, so the first term is 2/r = plus or minus 2/sqrt(3). | |
| 9 | B | Exponents, Logarithms & Radicals | Rewrite everything as a power of 5: the right side collapses to 5^(-12/x) and the left is 5^(-4), so x = 3. | |
| 10 | C | Basic Counting | Old plates number 26 times 10^4 and new ones 26^3 times 10^3; the ratio simplifies to 26^2 / 10. | |
| 11 | A | Coordinate Geometry | Dropping 15 units at slope 3 takes a run of 5, at slope 5 a run of 3, so the intercepts are 5 and 7. | |
| 12 | C | Linear Equations & Word Problems | Betty and Clare only matter as a combined sum (1000 - a) that doubles; the total 1500 then forces a = 400. | |
| 13 | E | Bases & Digits | The digit sum of a two-digit number is at most 18, so it must be 3 or 12; count numbers with those digit sums: 3 + 7. | |
| 14 | D | Exponents, Logarithms & Radicals | The exponent b must divide both 8 and 2, so b is at most 2; b = 2 gives a = 405 and sum 407. | |
| 15 | E | Games & Processes | Every match eliminates exactly one player and 99 players must be eliminated, so there are 99 matches regardless of byes; 99 = 9 * 11. | |
| 16 | E | Basic Counting | There are 2m * m * 3 = 6m^2 dinners and 365 days, so m^2 >= 61 and the least m is 8. | |
| 17 | B | Solid Geometry | Melted volume is 3/4 of 4/3 pi r^3 = pi r^3, and setting this equal to the cone's pi r^2 h / 3 gives h = 3r. | |
| 18 | D | Divisibility & Factors | For even n the factors are five consecutive odd integers, which always include multiples of 3 and 5 but not always 11, so 15. | |
| 19 | E | Circles | Join each intersection point of neighboring small semicircles to the two centers: the white region becomes two equilateral triangles plus sectors totaling 5/6 of a unit circle. | |
| 20 | D | Similar & Congruent Triangles | FG = 2 is parallel to AB = 5, so triangle EFG is a 2/5 copy of EAB; heights differ by 3, so the height is 5. | |
| 21 | C | Conditional Probability & States | All beads end up red exactly when both greens are drawn among the three pulls; track the count of greens and note each drawn green is replaced by red. | |
| 22 | B | Clocks, Calendars & Time | A full day has 180 chimes and the rest of Feb 26 has 91; 1912 = 10 * 180 + 112 lands on the eleventh day, March 9. | |
| 23 | D | Quadrilaterals & Polygon Areas | Joining the center O to the vertices, rectangle ABEF is four triangles each equal in area to one of the octagon's eight central triangles. | |
| 24 | E | Sequences & Series | The common difference is -2y, so xy = x - 3y and x/y = x - 5y; both give x(1 - y), forcing y = -3/5. | |
| 25 | B | Modular Arithmetic | The 90 candidates 1023, 1123, ..., 9923 step by 100, which is 1 mod 3, so they cycle through residues and exactly one third, 30, are multiples of 3. |
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.