2012 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.
The same key as a PDF 124 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 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. 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.