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2006 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2006 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 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 — answer key. 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.