2014 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 | C | Fractions & Decimals | Add the three fractions over the common denominator 10 to get 4/5, then the exponent -1 flips it to 5/4 before multiplying by 10. | |
| 2 | C | Linear Equations & Word Problems | The cat eats 7/12 of a can per day, so 6 cans last 10 full days with 2/12 left over, which runs out on the 11th morning. | |
| 3 | E | Linear Equations & Word Problems | Track the loaves in three batches (24, 16, 8) with prices 2.50, 1.25, 1.00 to get revenue 88, then subtract the cost 48 times 0.75 = 36. | |
| 4 | B | Arrangements with Restrictions | Place blue and yellow first: blue before yellow and not adjacent leaves only positions (1,3), (1,4), (2,4), and orange-then-red fills the rest uniquely. | |
| 5 | C | Statistics & Data | The cumulative percentages 10, 45, 75 show the middle student is in the 90 group, so the median is 90, while the weighted mean is 87. | |
| 6 | A | Ratios, Percents & Averages | Reduce to one cow for one day: b/(ac) gallons; then multiply by d cows and e days. | |
| 7 | B | Absolute Value & Inequalities | Only addition preserves both inequalities; a single negative example such as x = y = -2, a = b = -1 breaks II, III and IV at once. | |
| 8 | D | Number Properties | n!(n+1)! = (n!)^2 (n+1), so the expression is a square exactly when (n+1)/2 is a square; 18/2 = 9 works. | |
| 9 | C | Triangles: Area & Pythagorean | The third altitude goes to the hypotenuse, so compute the area two ways: (leg x leg) equals (hypotenuse x altitude). | |
| 10 | B | Ratios, Percents & Averages | The average of five consecutive integers is the middle one, so b = a + 2 and the new average is b + 2 = a + 4. | |
| 11 | C | Ratios, Percents & Averages | Coupon 1 beats coupon 2 when 0.1P > 20 (P > 200) and beats coupon 3 when 0.1P > 0.18(P - 100) (P < 225); only 219.95 fits. | |
| 12 | C | Circles | Each sector spans the hexagon's 120-degree interior angle, so the six sectors total two full circles of radius 3; subtract 18 pi from the hexagon area 54 root 3. | |
| 13 | C | Quadrilaterals & Polygon Areas | The hexagon splits into the triangle, three unit squares, and three gap triangles with unit legs and a 120-degree apex, each having the same area as the equilateral triangle. | |
| 14 | D | Coordinate Geometry | Intercepts p and -p make the origin the midpoint of PQ; the right angle at A puts A on the circle with diameter PQ, so p = OA = 10. | |
| 15 | C | Linear Equations & Word Problems | Compare the two plans for the remaining distance d - 35: at 35 mph versus 50 mph the times differ by exactly 1.5 hours. | |
| 16 | E | Quadrilaterals & Polygon Areas | The shaded region is a kite with diagonal HF of length 1 along the vertical midline and a horizontal diagonal of length 1/3, so its area is (1/2)(1)(1/3). | |
| 17 | D | Basic Probability | Only the largest die can equal the sum of the other two; pick which die it is (3 ways) and count ordered pairs adding to each possible total. | |
| 18 | B | Coordinate Geometry | A tilted square's vertices sit at heights 0, p, q, p+q, so p = 1 and q = 4, and each side is the hypotenuse of a 1-by-4 right triangle. | |
| 19 | A | Solid Geometry | XY joins opposite 3-D corners of the stack, (0,0,10) to (4,4,0); the part at heights 4 to 7 is 3/10 of its total length 2 root 33. | |
| 20 | D | Bases & Digits | Multiplying out 8 times 8, 88, 888, 8888 shows the product is 7, then k-2 ones, then 04, so the digit sum is k + 9. | |
| 21 | E | Diophantine Equations | Equal x-intercepts -5/a = -b/3 means ab = 15, so a runs over the divisors of 15 and the intercepts are -5, -5/3, -1, -1/3. | |
| 22 | E | Triangles: Area & Pythagorean | Reflect the 30-60-90 triangle to get tan 15 = 2 - sqrt 3; then CE = 20 - 10 sqrt 3, DE = 10 sqrt 3, and AE = 20. | |
| 23 | C | Transformations & Symmetry | The fold line makes a 60-degree angle with the base, so corner B lands exactly on corner D and the overlap is an equilateral triangle of area A/3. | |
| 24 | A | Sequences & Series | After n rounds, n(n+7)/2 numbers are listed and the count reaches n(n+4); n = 996 gives 499,494 listed ending at 996,000, so the answer is 996,000 + 506. | |
| 25 | B | Exponents, Logarithms & Radicals | Each gap between consecutive powers of 5 holds two or three powers of 2; counting 2013 powers of 2 over 867 gaps forces 279 triple gaps. |
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.