2009 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 | E | Linear Equations & Word Problems | Divide 128 by 12 and round up, because a partially used can still counts as a whole can. | |
| 2 | A | Number Properties | A total ending in 5 forces the number of pennies to be a multiple of 5, hence zero; then four coins of at least 5 cents total at least 20. | |
| 3 | C | Fractions & Decimals | Simplify from the innermost fraction outward: 1 + 1/2 = 3/2, its reciprocal is 2/3, and 1 + 2/3 = 5/3. | |
| 4 | A | Linear Equations & Word Problems | Budget time, not speed: the swim takes 1/8 hour and the run 1/2 hour, leaving 11/8 hour for the 15-mile ride. | |
| 5 | E | Bases & Digits | Repunit squares form a palindrome that climbs 1, 2, ..., 9 and back down, so the digits sum to 2(1+...+8)+9 = 81. | |
| 6 | A | Circles | The inscribed circle's diameter equals the semicircle's radius, so the semicircle has radius 4 and area 8 pi, exactly twice the circle's 4 pi. | |
| 7 | C | Ratios, Percents & Averages | Forty percent less than the whole-milk fat means 2 is 60% of it, so the whole-milk percentage is 2 divided by 0.6. | |
| 8 | B | Ratios, Percents & Averages | A senior ticket is 75% of full price, so full price is 6 / 0.75 = 8; then the six tickets cost 2(8) + 2(6) + 2(4). | |
| 9 | B | Sequences & Series | The third term a r^2 = 2009 = 7^2 * 41 and r > 1 is an integer, so r^2 | 2009 forces r = 7 and a = 41. | |
| 10 | B | Similar & Congruent Triangles | The altitude to the hypotenuse is the geometric mean of the pieces it cuts: BD^2 = 3 * 4, so BD = 2 sqrt 3 and the area is (1/2)(7)(BD). | |
| 11 | D | Algebraic Manipulation | The new box is s(s+1)(s-1) = s^3 - s by difference of squares, so the volume drops by exactly s; a drop of 5 means s = 5. | |
| 12 | C | Triangles: Area & Pythagorean | Triangle BCD forces BD > 17 - 5 = 12 and triangle ABD forces BD < 5 + 9 = 14, so the integer BD is 13. | |
| 13 | E | Exponents, Logarithms & Radicals | Write 12^(mn) = 2^(2mn) 3^(mn) and re-group each prime power as a power of P or Q: 2^(2mn) = P^(2n), 3^(mn) = Q^m. | |
| 14 | A | Quadrilaterals & Polygon Areas | The outer square has side a + b and the inner square side a - b, so an area ratio of 4 means a + b = 2(a - b). | |
| 15 | E | Sequences & Series | Each new outer square adds 4(n-1) diamonds, so F_n = 1 + 4(1 + 2 + ... + (n-1)) = 2n(n-1) + 1, and F_20 = 761. | |
| 16 | D | Absolute Value & Inequalities | Telescope a - d = (a-b) + (b-c) + (c-d) = ±2 ± 3 ± 4; the eight sign choices give |a-d| in {1, 3, 5, 9}, summing to 18. | |
| 17 | C | Similar & Congruent Triangles | EF is the hypotenuse of right triangle DEF with altitude DB = 5; triangles DBE and DBF are scaled copies of the 3-4-5 triangles DAB and DCB. | |
| 18 | D | Ratios, Percents & Averages | Take 100 children: 60 play soccer, 24 of them swim, so 36 soccer players don't swim, out of 70 non-swimmers: 36/70 = 51%. | |
| 19 | B | Divisibility & Factors | Rolling without slipping, the same point of B touches again only if r divides 100; 100 has 9 divisors, and r < 100 removes one. | |
| 20 | D | Linear Equations & Word Problems | The 1 km/min closing rate splits 3:1, so Lauren rides 1/4 km/min; after 5 minutes 15 km remain and Lauren alone needs 60 more minutes. | |
| 21 | C | Circles | The four small centers form a square of side 2r, so each is r sqrt 2 from the big center and the big radius is r(1 + sqrt 2). | |
| 22 | D | Basic Probability | By symmetry the two top numbers are a uniformly random pair of distinct labels from the bag; given the first, exactly 2 of the remaining 11 labels complete a 7. | |
| 23 | E | Similar & Congruent Triangles | Equal areas of AED and BEC force AB parallel to CD, so triangles AEB and CED are similar with ratio 9 : 12, and AE = (3/7)(14). | |
| 24 | C | Basic Probability | A plane through three vertices misses the interior only when all three lie on one face; 6 faces times C(4,3) = 24 bad triples out of C(8,3) = 56. | |
| 25 | B | Divisibility & Factors | I_k = 2^(k+2) 5^(k+2) + 2^6; factor out the smaller power of 2, and only the tie k = 4 gives an extra factor from 5^6 + 1. |
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.