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.
The same key as a PDF 125 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Fractions & Decimals | Add the three fractions first to get 4/5. The power -1 means flip that single fraction to 5/4, and only then multiply by 10. | |
| 2 | C | Linear Equations & Word Problems | One day uses 7/12 of a can. Ten days use 70/12 of the 72/12 in the box, so the last bit is eaten on the 11th morning. | |
| 3 | E | Linear Equations & Word Problems | Follow the loaves in three groups, 24, 16 and 8, to get 88 dollars of sales. Then subtract what all 48 loaves cost to make, 36 dollars. | |
| 4 | B | Arrangements with Restrictions | Place blue and yellow first, since they have the most rules. Only three spot pairs keep blue first with a gap; orange and red then fit in exactly one way. | |
| 5 | C | Statistics & Data | Pretend there are 100 students. The mean is 87. Counting up from the lowest score, the 50th student falls in the group that scored 90, so the median is 90. | |
| 6 | A | Ratios, Percents & Averages | First find how much one cow gives in one day, b/(ac) gallons. Then multiply by the number of cows d and the number of days e. | |
| 7 | B | Absolute Value & Inequalities | Adding two inequalities that point the same way is always safe. Subtracting, multiplying and dividing are not, and one example with negative numbers breaks them. | |
| 8 | D | Number Properties | The two factorials share a big common part, so each choice is a perfect square times (n+1)/2. Only 18/2 = 9 is a perfect square. | |
| 9 | C | Triangles: Area & Pythagorean | The missing altitude is the one drawn to the longest side. Find the area using the two legs, then use that same area with the longest side as the base. | |
| 10 | B | Ratios, Percents & Averages | The average of five consecutive whole numbers is the middle one. So b is a + 2, and the second list averages b + 2, which is a + 4. | |
| 11 | C | Ratios, Percents & Averages | Write each coupon's savings for a price P. Beating the 20-dollar coupon needs P above 200; beating the 18 percent coupon needs P below 225. Only 219.95 fits. | |
| 12 | C | Circles | Each sector opens as wide as the hexagon's corner, 120 degrees, which is one third of a circle. Six thirds make two whole circles of radius 3. | |
| 13 | C | Quadrilaterals & Polygon Areas | Cut the hexagon into the triangle, the three squares, and three thin gap triangles. Each gap triangle has two sides of length 1 with a 120-degree angle between them. | |
| 14 | D | Coordinate Geometry | Find the two slopes: one is m, the other is -1/m. Making the two y-intercepts add to zero gives m = 3, so the intercepts are 10 and -10. | |
| 15 | C | Linear Equations & Word Problems | Look only at the distance left after the first hour. Driving it at 35 instead of 50 miles per hour takes exactly one and a half hours longer. | |
| 16 | E | Quadrilaterals & Polygon Areas | Put the rectangle on a grid. The shaded shape is a kite with a tall diagonal of length 1 and a crossing diagonal of length 1/3. | |
| 17 | D | Basic Probability | Choose which die shows the total (3 ways). The other two dice must add up to at most 6, and there are 15 ordered ways to do that. | |
| 18 | B | Coordinate Geometry | Opposite corners of a square have heights adding to the same total, so 0 and 5 are opposite. Each side is the long side of a 1-by-4 right triangle. | |
| 19 | A | Solid Geometry | XY runs corner to corner through the stack and has length 2 root 33. The edge-3 cube covers heights 4 to 7, which is 3/10 of the drop. | |
| 20 | D | Bases & Digits | Work out 8 times 88, 888, 8888. The answer is always a 7, then a run of ones, then 04, so the digit sum is k + 9. | |
| 21 | E | Diophantine Equations | Setting the two x-intercepts equal gives a times b = 15. So a is 1, 3, 5 or 15, and the crossing points are -5, -5/3, -1 and -1/3. | |
| 22 | E | Triangles: Area & Pythagorean | Draw a 30-degree line from B; BE cuts that angle in half. The angle bisector rule gives CE, then DE is 10 root 3 and AE is 20. | |
| 23 | C | Transformations & Symmetry | The fold line is tilted at 60 degrees, so the folded corner lands exactly on the opposite corner. The doubled part is an equilateral triangle, one third of the paper. | |
| 24 | A | Sequences & Series | The 500,000th listed number is 500,000 plus how many numbers were skipped before it. Up to that point, 1 + 2 + ... + 996 = 496,506 numbers were skipped. | |
| 25 | B | Exponents, Logarithms & Radicals | Since 5 is between 4 and 8, each gap between neighbouring powers of 5 holds two or three powers of 2. Share 2013 powers of 2 among 867 gaps. |
2014 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.