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2009 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2009 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | E | Linear Equations & Word Problems — Ten cans hold only 120 ounces, which is 8 short, so one more can is needed even though it will not be full. |
| 2 | A | Number Properties — Every choice is a multiple of 5, so the pennies must total a multiple of 5, which means none; four coins of at least 5 cents make at least 20. |
| 3 | C | Fractions & Decimals — Work from the innermost part outward: 1 + 1 = 2, then 1 + 1/2 = 3/2, flip it to get 2/3, then 1 + 2/3 = 5/3. |
| 4 | A | Linear Equations & Word Problems — Find how long the swim and the run take, subtract from 2 hours, and use the time that is left for the 15-mile bike ride. |
| 5 | E | Bases & Digits — Squaring 11, 111 and 1111 gives 121, 12321 and 1234321; the digits climb to the number of ones and come back down, with no carrying up to nine ones. |
| 6 | A | Circles — The small circle reaches from the middle of the flat edge up to the top of the arc, so the semicircle's radius is the small circle's diameter, 4. |
| 7 | C | Ratios, Percents & Averages — Taking 40% away from the whole-milk amount leaves 60% of it, so 60% of the whole-milk amount is 2, and the whole-milk amount is 2 divided by 0.6. |
| 8 | B | Ratios, Percents & Averages — A senior ticket is 75% of the full price, so the full price is 6 divided by 0.75, which is 8 dollars; a child ticket is half of that. |
| 9 | B | Sequences & Series — The ratio squared must divide 2009 = 7 x 7 x 41, so the ratio is 7 and a = 2009 / 49 = 41. |
| 10 | B | Similar & Congruent Triangles — The height BD is a leg of two small right triangles. Using the Pythagorean theorem in all three triangles gives BD squared = 3 x 4 = 12. |
| 11 | D | Algebraic Manipulation — The new box has volume s(s+1)(s-1), which is s cubed minus s, so it is always s less than the cube. A drop of 5 means s = 5. |
| 12 | C | Triangles: Area & Pythagorean — In triangle BCD, BD + 5 must be more than 17, so BD > 12. In triangle ABD, BD < 5 + 9 = 14. So BD = 13. |
| 13 | E | Exponents, Logarithms & Radicals — Split 12 into 2 x 2 x 3, so 12^(mn) = 2^(2mn) times 3^(mn). Then 2^(2mn) = P^(2n) and 3^(mn) = Q^m. |
| 14 | A | Quadrilaterals & Polygon Areas — The outer square's side is a + b and the inner square's is a - b. Four times the area means twice the side. |
| 15 | E | Sequences & Series — Each new ring with n diamonds per side holds 4(n - 1) diamonds, so F_20 = 1 + 4(1 + 2 + ... + 19) = 761. |
| 16 | D | Absolute Value & Inequalities — Put a at 0 and step 2, then 3, then 4, each to the left or right. The possible distances from the start are 1, 3, 5 and 9. |
| 17 | C | Similar & Congruent Triangles — DB is the height of right triangle DEF to its long side EF; the triangles DBE and DBF are enlarged copies of the 3-4-5 triangles DAB and DCB. |
| 18 | D | Ratios, Percents & Averages — Pretend there are 100 children: 60 play soccer, 24 of those swim, so 36 soccer players do not swim, out of 70 children who do not swim. |
| 19 | B | Divisibility & Factors — Rolling without slipping, circle B has turned 100/r times when it gets back to the start; the same point is touching again only if r divides 100. |
| 20 | D | Linear Equations & Word Problems — The gap closes 1 km per minute, split 3 to 1, so Lauren rides 1/4 km per minute. After 5 minutes, she alone rides the last 15 km. |
| 21 | C | Circles — The four small centers form a square of side 2r. The big center is half a diagonal, r times root 2, from each, so R = r(1 + root 2). |
| 22 | D | Basic Probability — Random stickers make the first top number any of the 12, and the second any of the other 11. Exactly 2 of those 11 make 7. |
| 23 | E | Similar & Congruent Triangles — Adding triangle ABE to both equal areas shows ABD and ABC have equal area, so AB is parallel to CD and AE : EC = 9 : 12. |
| 24 | C | Basic Probability — Count the bad choices: the plane misses the inside only when all three corners lie on one face. That is 6 faces times 4 choices, 24 out of 56. |
| 25 | B | Divisibility & Factors — I_k is 10 to the power k+2, plus 64. Pull out the smaller power of 2 and an odd number remains, except at the tie k = 4. |
2009 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.