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.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | D | Linear Equations & Word Problems | Convert each frosting time to a rate per minute (3 and 2 cupcakes per minute), add the rates, and multiply by 5 minutes. | |
| 2 | E | Quadrilaterals & Polygon Areas | Cutting a square in half along a midline halves one side and leaves the other side untouched: 4 by 8. | |
| 3 | E | Absolute Value & Inequalities | Distance traveled is the sum of the absolute differences of consecutive positions: 4 units left, then 11 units right, total 15. | |
| 4 | C | Angles & Polygons | Rays BC and BD can lie on the same side of ray BA or on opposite sides; the same side gives 24 - 20 = 4, the smaller option. | |
| 5 | B | Linear Equations & Word Problems | Half of 100 is 50 females, half of those is 25 litters, and 25 litters averaging 4 kittens is 100 kittens; add the 100 adults. | |
| 6 | D | Systems of Equations | The reciprocal condition 1/x = 4/y just says y = 4x; substitute into xy = 9 to get x = 3/2, y = 6. | |
| 7 | C | Ratios, Percents & Averages | Take 5 marbles: 3 blue, 2 red; doubling red gives 4 red out of 7 total. | |
| 8 | D | Systems of Equations | Adding all three pair sums counts each number twice, so the total is 24; subtracting each pair sum from 24 recovers the missing number. | |
| 9 | D | Basic Probability | Each die shows one of three values with equal probability, so there are 9 equally likely value pairs, and exactly three of them (2+5, 4+3, 6+1) sum to 7. | |
| 10 | C | Sequences & Series | The 12 angles sum to 360, so 2a + 11d = 60; d must be even, and the largest even d with a positive is 4, giving a = 8. | |
| 11 | D | Circles | Radii to the tangent points are parallel, so the right triangles from C to each center are similar with ratio 5 : 3, and AC = BC + 8. | |
| 12 | A | Clocks, Calendars & Time | Every year shifts the weekday by one, plus one per leap day; 200 years and 49 leap days give 249, which is 4 mod 7: back 4 from Tuesday. | |
| 13 | C | Statistics & Data | The result is a weighted average with weights 1/16, 1/16, 1/8, 1/4, 1/2 in entry order; put the largest number last to maximize and first to minimize. | |
| 14 | B | Paths & Grids | Color depends only on the parity of row + column; with corners black, black squares are those with an even coordinate sum, one more than half of 961. | |
| 15 | B | Triangles: Area & Pythagorean | AB is a unit base, so only the height of C is needed; the two segments have slopes -1/2 and 2, and they meet 2/5 below AB. | |
| 16 | C | Linear Equations & Word Problems | Two runners coincide when their speed difference times t is a whole number of laps; all three meet at the lcm of the pairwise meeting times. | |
| 17 | C | Algebraic Manipulation | Cancel one factor of a - b, write a^2 + ab + b^2 as (a-b)^2 + 3ab, and get 9ab = 70(a-b)^2; coprimality forces (a-b)^2 to divide 9. | |
| 18 | E | Circles | Each arc has radius 1 and spans 120 degrees; three vertices carry 240-degree outward bulges and three carry 120-degree dents, so hexagon plus 2pi minus pi. | |
| 19 | D | Linear Equations & Word Problems | Tuesday plus Wednesday also paints 50%, so comparing with Monday gives the helpers-to-Paula rate ratio; dividing Tuesday by Wednesday then leaves only the lunch break. | |
| 20 | A | Basic Probability | The rotation splits the grid into the center and two 4-cycles; a square stays white only if its predecessor was white, so each 4-cycle needs no adjacent whites. | |
| 21 | C | Coordinate Geometry | EF and GH are midlines parallel to AD, FG is parallel to BC, and AD is perpendicular to BC, so EFGH is a rectangle with sides AD/2 and BC/2. | |
| 22 | A | Diophantine Equations | The sums are m^2 and n^2 + n; multiply by 4 and complete the square to get (2m)^2 - (2n+1)^2 = 847, then factor 847 = 7 * 11^2. | |
| 23 | B | Basic Counting | Everyone has the same number k of friends, 1 to 4; count k = 1 (pairings) and k = 2 (cycle covers), and complements give k = 3, 4. | |
| 24 | E | Algebraic Manipulation | Adding the equations gives (a-b)^2 + (b-c)^2 + (a-c)^2 = 14, forcing the gaps a-b and b-c to be 1 and 2 in some order. | |
| 25 | D | Geometric Probability | Order x < y < z, then shift y down 1 and z down 2; the good region becomes all ordered triples in [0, n-2], giving probability ((n-2)/n)^3. |
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.