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2012 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2012 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | D | Linear Equations & Word Problems — Work out how many cupcakes each person frosts in the 5 minutes on their own, then add the two counts. |
| 2 | E | Quadrilaterals & Polygon Areas — A cut straight through the middle splits one side of 8 into two 4s and leaves the other side at 8. |
| 3 | E | Absolute Value & Inequalities — Split the trip at the turning point. Find the length of each part (4 units and 11 units) and add them. |
| 4 | C | Angles & Polygons — Rays BC and BD can be on the same side of BA or on opposite sides. The same side gives 24 - 20 = 4, the smaller choice. |
| 5 | B | Linear Equations & Word Problems — Follow the halves one at a time: 50 females, 25 litters, 25 times 4 = 100 kittens, then add the 100 adult cats. |
| 6 | D | Systems of Equations — The reciprocal of the smaller number is 4 times the reciprocal of the larger, so the larger number is 4 times the smaller. Put that into the product 9. |
| 7 | C | Ratios, Percents & Averages — Pretend the bag holds 5 marbles: 3 blue and 2 red. Doubling the red gives 4 red, and the bag now holds 7 marbles. |
| 8 | D | Systems of Equations — Add the three pair sums. Each number is counted twice, so the three numbers total 24. Take away each pair sum from 24 to find the number left out. |
| 9 | D | Basic Probability — Each die really has three equally likely values. Whatever the even die shows, exactly one of the three odd values makes the total 7. |
| 10 | C | Sequences & Series — The 12 angles add to 360, so 2a + 11d = 60. A bigger step d means a smaller first angle; d must be even, so try d = 4. |
| 11 | D | Circles — The radii to the tangent line both meet it at right angles, so the two right triangles with a corner at C are similar, in the ratio 5 to 3. |
| 12 | A | Clocks, Calendars & Time — Each year moves the weekday by 1, each leap day by 1 more: 200 + 49 = 249 days, 4 past whole weeks. Go back 4 from Tuesday. |
| 13 | C | Statistics & Data — Each averaging step halves the weight of the earlier numbers, so the last number counts most. Put 5 last for the largest result and 1 last for the smallest. |
| 14 | B | Paths & Grids — A row of 31 starting black has 16 black squares; one starting red has 15. The 31 rows alternate, starting black, so 16 rows have 16 blacks. |
| 15 | B | Triangles: Area & Pythagorean — AB has length 1, so only the height of C below AB matters. One segment drops 1 per 2 across, the other 2 per 1 across; they meet 2/5 down. |
| 16 | C | Linear Equations & Word Problems — Two runners are together whenever the faster has gained a whole lap, 500 m. Find when each pair meets, then the first time on both lists. |
| 17 | C | Algebraic Manipulation — Cancel a - b and call it d, giving 9ab = 70d^2. Since a and b share no factor, d shares none with ab, so d divides 3. |
| 18 | E | Circles — Each arc has radius 1 and turns 120 degrees. Take the hexagon, add three outside bumps of 240 degrees, and remove three inside dents of 120 degrees. |
| 19 | D | Linear Equations & Word Problems — Tuesday's 24% plus Wednesday's 26% equals Monday's 50%. Comparing them gives the helpers' speed compared with Paula's, and then only the lunch break is unknown. |
| 20 | A | Basic Probability — A spot stays white only if it and the square rotating into it were both white. Corners and edges each cycle in fours; no two neighbours may be white. |
| 21 | C | Coordinate Geometry — EF and GH are half of AD and point straight up; FG and EH are half of BC and lie flat. So EFGH is a rectangle. |
| 22 | A | Diophantine Equations — The sums are m^2 and n(n + 1). Multiply by 4 to complete the square, giving a difference of two squares equal to 847 = 7 x 11 x 11. |
| 23 | B | Basic Counting — Everyone has 1 to 4 friends. Swapping friends and non-friends turns 1 into 4 and 2 into 3, so count only 1 and 2 friends, then double. |
| 24 | E | Algebraic Manipulation — Add the equations: the sum is (a-b)^2 + (b-c)^2 + (a-c)^2 = 14, so the gaps a - b and b - c are 1 and 2. |
| 25 | D | Geometric Probability — Sort so x < y < z, then slide y down 1 and z down 2. The gaps disappear, and the probability is ((n - 2)/n)^3. |
2012 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.