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2011 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2011 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | D | Linear Equations & Word Problems — Split the bill into three parts. Only the extra half hour costs money, and half an hour is 30 minutes at 10 cents each. |
| 2 | E | Number Properties — Fourteen small bottles hold only 490 milliliters, which is not quite 500, so she needs one more bottle. |
| 3 | D | Functions — Work from the inside out. The inner brackets are 2/3 and 1/2; then average those two numbers together with 0. |
| 4 | A | Sequences & Series — The two sums share every even number from 12 to 100. Those cancel when you subtract, leaving just 102 minus 10. |
| 5 | C | Ratios, Percents & Averages — The grade sizes are in the ratio 4 : 2 : 1, so pretend there are 4, 2 and 1 students. Add up all their minutes and divide by 7. |
| 6 | C | Sets, Estimation & Miscellaneous — The union always contains all 20 elements of A. It has exactly 20 when every element of B is already in A. |
| 7 | B | Absolute Value & Inequalities — An absolute value is never negative, so |−3x| + 5 is always at least 5 and can never equal 0. Each other equation has an actual solution. |
| 8 | C | Ratios, Percents & Averages — Pretend there are 100 birds. Leaving out the 25 swans leaves 75 birds, and 30 of them are geese, so the answer is 30 out of 75, which is 40%. |
| 9 | A | Coordinate Geometry — The lines x = −c and x = d sit on opposite sides of the y-axis, so the width is c + d. Likewise the height is a + b. |
| 10 | B | Divisibility & Factors — Factor 1771 cents as 7 × 11 × 23. A majority of 30 is 16 to 30 students, so 23 bought; each bought 7 pencils at 11 cents. |
| 11 | B | Triangles: Area & Pythagorean — Make the side 8, so AE = 7 and EB = 1. Each corner triangle has legs 7 and 1; the inner square has area 49 + 1 = 50. |
| 12 | A | Linear Equations & Word Problems — Call the number of two-point shots y. They give 2y points, three-pointers also give 2y points, and free throws give y + 1 points, so 5y + 1 = 61. |
| 13 | A | Basic Counting — The first digit must be 2 or 5 and the last digit must be 2 or 8. If the first digit is 2, the last digit can only be 8. |
| 14 | B | Basic Probability — With diameter d, the area is less than the circumference exactly when d is less than 4. So the dice must sum to 2 or 3: 3 of 36 rolls. |
| 15 | C | Linear Equations & Word Problems — Miles per gallon for the whole trip is total miles divided by total gallons. Only the miles after the first 40 use any gas. |
| 16 | B | Exponents, Logarithms & Radicals — Square the sum. The two roots multiply to the square root of 81 − 72 = 9, which is 3, so the square is 18 + 6 = 24. |
| 17 | C | Sequences & Series — Two overlapping groups of three share two terms, so each term equals the term three places later. So H equals B, and A + B = 30 − 5. |
| 18 | C | Circles — Circle C passes through the point where A and B touch, so it overlaps each of them in a lens of area π/2 − 1. What is left is 2. |
| 19 | E | Number Properties — Call 1991 a² and 2001 b² + 9. Then b² − a² = 141, so a = 70 or a = 22. Only 22 also makes 2011 a perfect square. |
| 20 | D | Geometric Probability — A chord as long as the radius covers 60 degrees. Fix the first point; the chords cross only if the second point lands within 60 degrees of it, either side. |
| 21 | D | Conditional Probability & States — The two pairs weigh the same only when both are all real coins or each holds exactly one fake. Count those two kinds of outcomes and compare. |
| 22 | C | Arrangements with Restrictions — The five diagonals join up into one closed loop A, C, E, B, D, back to A. So count colorings where neighbours on this loop differ. |
| 23 | C | Games & Processes — Each student gets an evenly spaced list. The numbers she skips start one step later and are spaced three times as far apart: 2, 5, 14, 41, 122, 365. |
| 24 | D | Solid Geometry — Each tetrahedron has volume 1/3. The other tetrahedron's faces cut off its four corners, each a half-size copy of volume 1/24, leaving 1/3 − 4/24 = 1/6. |
| 25 | C | Triangles: Area & Pythagorean — The rays must pass through all four corners. That forces both coordinates of the point to be multiples of 2/n, so the question becomes counting grid points. |
2011 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.