2012 AMC 10B
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 132 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Linear Equations & Word Problems | Each classroom has 16 more students than rabbits. There are 4 classrooms, so there are 4 times 16, which is 64, more in all. | |
| 2 | E | Quadrilaterals & Polygon Areas | The circle touches the top and bottom of the rectangle, so the width equals the circle's diameter, 10. The length is twice that, 20. | |
| 3 | B | Coordinate Geometry | The mirror line y = 2000 is flat, so the x-coordinate stays 1000. The point is 12 above the line, so its reflection is 12 below it. | |
| 4 | A | Modular Arithmetic | Put the two leftover piles together: 4 + 3 = 7 marbles. Seven marbles fill one more bag of 6, leaving 1. | |
| 5 | D | Ratios, Percents & Averages | The tax and the tip are both taken from the meal price, so every dollar of meal costs 1.25 dollars in total. Divide 27.50 by 1.25. | |
| 6 | A | Absolute Value & Inequalities | Making the first number bigger and the subtracted number smaller both make the difference bigger. The two roundings add up instead of cancelling. | |
| 7 | D | Linear Equations & Word Problems | Both animals hid the same total. Write the chipmunk's total as 3 times its holes and the squirrel's as 4 times 4 fewer holes, and set them equal. | |
| 8 | B | Absolute Value & Inequalities | The square must be 4, 9 or 16, so x - 2 is 2, 3 or 4 or their negatives. The six answers pair up around 2. | |
| 9 | A | Number Properties | Split the totals into three pairs that add to 26, 15 and 16. Only the pair adding to 15, an odd number, is forced to contain an odd integer. | |
| 10 | D | Divisibility & Factors | Cross-multiply to get M times N = 36. Every divisor of 36 chosen as M gives exactly one N, and 36 has 9 divisors. | |
| 11 | A | Arrangements with Restrictions | Fill in Friday (cake) first, then work outward from it. Every other day then has 3 choices, since only the neighbor already filled in is ruled out. | |
| 12 | B | Triangles: Area & Pythagorean | The 45-degree angle makes triangle ABC a right triangle with two equal legs of 10. Then AD is the long side of a right triangle with legs 10 and 30. | |
| 13 | B | Linear Equations & Word Problems | Speeds add, not times. Pretend the escalator has 120 steps: Clea walks 2 steps per second, and with the escalator she moves 5, so the escalator alone moves 3. | |
| 14 | D | Quadrilaterals & Polygon Areas | Each triangle is 3 tall, so the two tips are 6 - 2√3 apart. The rhombus has a 60-degree angle at each tip, so it is two equilateral triangles. | |
| 15 | D | Games & Processes | There are 15 games, so 15 wins in total. Six teams can't share 15 wins equally, but five teams can each win 3 if they all beat the sixth team. | |
| 16 | A | Circles | Connect the three centers to make an equilateral triangle of side 4. The triangle is the middle gap plus three 60-degree slices of circle that together make half a circle. | |
| 17 | C | Solid Geometry | When a sector rolls into a cone, its curved edge becomes the base circle and its radius 12 becomes the slant side. That gives base radii 4 and 8. | |
| 18 | C | Conditional Probability & States | Imagine 50,000 people: 100 are sick and test positive, but 2% of the 49,900 healthy people, 998 people, also test positive. So about 1 in 11 positives is real. | |
| 19 | C | Quadrilaterals & Polygon Areas | BFDG is a trapezoid: BF and GD are both vertical, 6 apart. Similar triangles give BF = 30/4 = 7.5, and GD is half of 30, so 15. | |
| 20 | A | Games & Processes | Silvia only adds 50, so Bernardo wins exactly when he hands her a number from 950 to 999. Find the smallest start that lands there. | |
| 21 | A | Triangles: Area & Pythagorean | Two points 2a apart, each exactly a from a third point, force that third point to be the midpoint. The fourth point then makes an equilateral triangle. | |
| 22 | B | Arrangements with Restrictions | The numbers written so far always form a run of consecutive numbers. Reading the list backward, each number removed must be an end of the run: 2 choices, nine times. | |
| 23 | D | Solid Geometry | The line from the removed corner to the opposite corner stands straight up from the cut face. Take the cut-off corner's height, 1/√3, away from √3. | |
| 24 | B | Basic Counting | Give each song a label: the group of girls who like it. There are 7 allowed labels, and the three two-girl labels must each be used at least once. | |
| 25 | E | Paths & Grids | After the bug crosses an arrow in the 1st, 3rd, 5th or 7th column it can never go back. So count routes one stage at a time and multiply. |
2012 AMC 10B. 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.