2006 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 134 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | A | Linear Equations & Word Problems | Find the cost of the sandwiches and the cost of the sodas separately, then add the two amounts. | |
| 2 | C | Functions | Do the inside operation first. Then use its result as the second input of the outside operation, keeping it inside parentheses. | |
| 3 | B | Ratios, Percents & Averages | Think of the ages as 3 equal parts and 5 equal parts. Alice's 5 parts are 30 years, so one part is 6 years. | |
| 4 | E | Bases & Digits | Make the hour digits and the minute digits as big as possible separately. The hour 9 beats 10, 11 and 12, and the minutes can be 59. | |
| 5 | D | Linear Equations & Word Problems | Give each slice its own price: every slice costs 1 dollar, and the 2-dollar anchovy charge is shared only by the 4 anchovy slices. | |
| 6 | B | Exponents, Logarithms & Radicals | Write the left side as a 7th power too: (7x)^14 = (49x^2)^7. Equal 7th powers mean equal bases, so 49x^2 = 14x and x = 2/7. | |
| 7 | A | Quadrilaterals & Polygon Areas | Moving pieces does not change area, so the square has area 144 and side 12. The long edge 18 − y of each hexagon becomes a side of the square. | |
| 8 | E | Quadratics | Plug both points into y = x^2 + bx + c and subtract the two equations. The c cancels, giving b = −6, and then c = 11. | |
| 9 | C | Number Properties | Sum = (how many numbers) times (their average). Try runs of 2, 3, 4, 5, 6 numbers and see which give 15 using only positive numbers. | |
| 10 | E | Number Properties | Call the integer k. Then the square root of x equals 120 − k^2, which cannot be negative, so k can be 0, 1, 2, up to 10: eleven values. | |
| 11 | C | Coordinate Geometry | Expand the left side. The x^2 and y^2 cancel, leaving xy = 0, which is true when x = 0 or when y = 0: two lines, the two axes. | |
| 12 | C | Circles | Both drawings give the same half-circle of radius 8. In II the rope reaches past the nearby corner with 4 feet left over, adding a quarter-circle of radius 4. | |
| 13 | D | Expected Value | Winning needs an even first roll (chance 1/2), then a match (chance 1/6). So the chance is 1/12, and the fair prize is 12 × 5 = 60 dollars. | |
| 14 | B | Sequences & Series | Each ring hangs on the one above, so the holes line up end to end. Add the holes, 18 down to 1, plus 1 cm at each end. | |
| 15 | D | Linear Equations & Word Problems | Both runners take the same time for one lap, 2π/5 minutes. Running toward each other, they meet twice per lap time, once every π/5 minutes. | |
| 16 | D | Similar & Congruent Triangles | Both circles sit in the corner at A, so the big one is twice as far from A. Centers 3 apart put the small center 3 from A. | |
| 17 | A | Quadrilaterals & Polygon Areas | Put the rectangle on a grid. Every drawn line has slope 1 or −1, so WXYZ is a square whose two diagonals both have length 1. | |
| 18 | C | Basic Counting | Glue the two letters into one block. It can start in 5 of the 6 spots; fill it with 26 × 26 letter pairs and the rest with 10^4 digits. | |
| 19 | C | Angles & Polygons | Three evenly spaced angles adding to 180 have middle angle 60. Write them as 60 − d, 60, 60 + d, with d from 1 to 59. | |
| 20 | E | Modular Arithmetic | Two numbers differ by a multiple of 5 exactly when they leave the same remainder when divided by 5. Six numbers but only five remainders, so it always happens. | |
| 21 | E | Basic Counting | Count the opposite: four-digit numbers with no 2 and no 3. There are 7 × 8 × 8 × 8 = 3584 of them; subtract from 9000. | |
| 22 | C | Diophantine Equations | Every payment is made of 300s and 210s, added or subtracted, so it is a multiple of 30. Three goats minus two pigs makes exactly 30. | |
| 23 | B | Circles | Radii to C and D meet the tangent at right angles, so triangles ACE and BDE have the same shape, scaled 3 to 8. Also CE = 4. | |
| 24 | B | Solid Geometry | Cut the octahedron into two pyramids with a square base halfway up the cube. That square's diagonal is 1, so its area is 1/2, and each pyramid is 1/2 tall. | |
| 25 | C | Basic Probability | By symmetry fix the first two moves (3 × 2 ways). Only 3 ways to finish visit every corner: 18 good paths out of 3^7 = 2187. |
2006 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.