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.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Linear Equations & Word Problems | Each classroom has 18 - 2 = 16 more students than rabbits, so four classrooms have 4 * 16 = 64 more. | |
| 2 | E | Quadrilaterals & Polygon Areas | The inscribed circle touches both long sides, so the rectangle's width is the diameter 10; the length is twice that. | |
| 3 | B | Coordinate Geometry | Reflecting over a horizontal line keeps x and sends y to 2(2000) - y, so 2012 goes to 1988. | |
| 4 | A | Modular Arithmetic | Remainders add: 4 + 3 = 7 marbles left over combine to fill one more bag, leaving 7 - 6 = 1. | |
| 5 | D | Ratios, Percents & Averages | Tax and tip are both percentages of the same base price, so the total is 1.25 times the meal: 27.50 / 1.25 = 22. | |
| 6 | A | Absolute Value & Inequalities | Rounding x up and y down both push the difference the same way, so the estimate is x - y + 2e, strictly larger. | |
| 7 | D | Linear Equations & Word Problems | Equal acorn totals give 3h = 4(h - 4): the four missing holes cost the squirrel 16 acorns, which its extra acorn per hole must make up. | |
| 8 | B | Absolute Value & Inequalities | (x-2)^2 must be 4, 9, or 16, so x - 2 is ±2, ±3, ±4; the six values are symmetric about 2 and sum to 12. | |
| 9 | A | Number Properties | An even sum needs no odd addends but an odd sum needs one; the middle pair adds to 15, so exactly one odd integer is forced. | |
| 10 | D | Divisibility & Factors | Cross-multiplying gives MN = 36, so ordered pairs correspond to divisors of 36 = 2^2 * 3^2, which number (2+1)(2+1) = 9. | |
| 11 | A | Arrangements with Restrictions | Fix Friday as cake first; every other day then has 3 options once its neighbor on the Friday side is chosen, giving 3^6. | |
| 12 | B | Triangles: Area & Pythagorean | The 45-degree angle makes ABC an isosceles right triangle with legs 10, so AD is the hypotenuse of a 10-by-30 right triangle, sqrt(1000). | |
| 13 | B | Linear Equations & Word Problems | Rates add: walking covers 1/60 of the escalator per second and walking-plus-riding 1/24, so riding alone covers 1/24 - 1/60 = 1/40 per second. | |
| 14 | D | Quadrilaterals & Polygon Areas | Each triangle has height 3, so the apexes are 6 - 2√3 apart; the side lines cross at height √3, making the other diagonal 2√3 - 2. | |
| 15 | D | Games & Processes | Fifteen wins cannot split evenly among six teams, but five teams with 3 wins each works: all beat a winless sixth and split their own games evenly. | |
| 16 | A | Circles | Join the centers: the side-4 equilateral triangle equals the middle region plus three 60-degree sectors, so total = three circles + triangle - half a circle. | |
| 17 | C | Solid Geometry | Each sector's arc becomes the base circumference and the radius 12 becomes the slant height; base radii 4 and 8 give heights 8√2 and 4√5. | |
| 18 | C | Conditional Probability & States | Picture 500 people: 1 true positive and about 0.02 * 499 ≈ 10 false positives, so roughly 1 in 11 positive tests is real. | |
| 19 | C | Quadrilaterals & Polygon Areas | BFDG is a trapezoid with parallel sides BF and GD; similar triangles EBF and EAD give BF = 7.5, and GD = 15 with height 6. | |
| 20 | A | Games & Processes | Bernardo wins exactly when one of his outputs 2N, 4N+100, 8N+300, 16N+700 lies in [950, 999], because Silvia adds only 50; the smallest N is 16. | |
| 21 | A | Triangles: Area & Pythagorean | Two length-a segments joining the ends of the 2a segment to one point force it to be the midpoint; the fourth point then completes an equilateral triangle. | |
| 22 | B | Arrangements with Restrictions | The numbers placed so far always form a block of consecutive integers, so reading backwards each removed number is an end of the block: 2 choices, nine times. | |
| 23 | D | Solid Geometry | The cut face is perpendicular to the space diagonal from the removed corner; that corner's height above the face is 1/√3, so √3 - 1/√3 remains. | |
| 24 | B | Basic Counting | Label each song by the set of girls liking it (7 options, never all three); the three pair-labels must all appear, so patterns are 1-1-1-other or 2-1-1. | |
| 25 | E | Paths & Grids | Once the bug exits an arrow in columns 1, 3, 5, 7 it can never go back, so the count multiplies stage by stage: 1, 5, 120, 1200 per arrow. |
Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic tags, difficulty ratings and key insights on this page are original to this site. No problem statements are reproduced here — each links to its own page.