Printable paper
2005 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2005 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | D | Ratios, Percents & Averages — The same 2 dollar tip is a smaller percent of a bigger bill. Work backward from each tip to each whole bill, then subtract. |
| 2 | C | Functions — Do the inner operation first: 1 star 2 is -3. Then -3 star 3 has a top of 0 and a bottom of -6, so the answer is 0. |
| 3 | B | Linear Equations & Word Problems — The first equation has no b, so solve it to get x = -2. Put that x into the second equation and solve for b. |
| 4 | B | Triangles: Area & Pythagorean — With width w and length 2w, the Pythagorean theorem gives x squared = 5 w squared. The area 2 w squared is two fifths of that. |
| 5 | A | Linear Equations & Word Problems — Each group of 5 windows gets one free. Alone, 7 and 8 each earn one free window; together, 15 earns three. Buying together gains one extra free window. |
| 6 | B | Statistics & Data — Turn each average into a total: 20 times 30 is 600 and 30 times 20 is 600. All 50 numbers add to 1200, so the average is 24. |
| 7 | B | Linear Equations & Word Problems — Josh rides twice as long at four fifths the speed, so he covers 8 fifths of Mike's distance. Mike's 5 parts plus Josh's 8 parts make 13 miles. |
| 8 | C | Triangles: Area & Pythagorean — The corner triangle on side AB has legs 1 and (1 + inner side). The Pythagorean theorem with 50 gives the inner side 6. |
| 9 | B | Basic Probability — Only the spots of the two O tiles matter. There are 10 ways to pick 2 spots out of 5, and XOXOX is exactly one of them. |
| 10 | A | Quadratics — Combine ax and 8x into (a + 8)x. One solution means the discriminant is zero, so (a + 8) squared is 144 and a + 8 is 12 or -12. |
| 11 | B | Solid Geometry — Red faces are the outside of the big cube: 6 n squared. All faces: 6 n cubed. The fraction red is 1 over n, so n = 4. |
| 12 | B | Circles — Each bulge is a sector minus a small triangle, and the four small triangles fill the big one. So the trefoil is just four 60 degree sectors. |
| 13 | E | Exponents, Logarithms & Radicals — Every exponent is a multiple of 50. Take the 50th root of each part to get 130n > n squared > 16, which means n is from 5 to 129. |
| 14 | E | Bases & Digits — The middle digit is decided by the other two, and it is a whole number only when the first and last digits are both odd or both even. |
| 15 | E | Divisibility & Factors — Write the product as 2^8 times 3^4 times 5^2 times 7. A cube divisor uses each prime a multiple of 3 times, giving 3 times 2 choices. |
| 16 | D | Bases & Digits — A two-digit number minus its digit sum is 9 times the tens digit. Only 36 ends in 6, so the tens digit is 4 and the units digit is free. |
| 17 | D | Sequences & Series — Each letter is on exactly two segments, so the five sums add to twice 30, which is 60. The middle of five evenly spaced numbers is their average, 12. |
| 18 | A | Conditional Probability & States — List every way the series can go when B wins game 2 and A wins the series. Weight shorter series more, since each game is a coin flip. |
| 19 | D | Triangles: Area & Pythagorean — The tilted square rests on its neighbors' inner top corners, at height 1. Its bottom tip is half an inch lower, and B is one diagonal above the tip. |
| 20 | A | Quadrilaterals & Polygon Areas — Every angle is 135 degrees, so the octagon is a 2 by 2 square with four small corners cut off. Each corner is a right triangle with legs one half. |
| 21 | B | Divisibility & Factors — The sum is n(n+1)/2. Dividing 6n by it leaves 12 over (n+1), so n+1 must be a divisor of 12 bigger than 1. |
| 22 | D | GCD & LCM — Common numbers are multiples of 12. S stops at 4 times 2005 = 8020, which is lower than where T stops, so count the multiples of 12 up to 8020. |
| 23 | C | Circles — O is the midpoint of DE, so triangle DCE is twice triangle DCO. Triangles DCO and ABD share the height DC, so compare their bases CO and AB. |
| 24 | B | Primes — The largest prime factor equals the square root only for a prime squared. So look for two prime squares 48 apart: 121 and 169. |
| 25 | D | Triangles: Area & Pythagorean — Triangles ADE and ABC share the angle at A, so their areas compare as 19 times 14 against 25 times 42, which is 19 to 75. The rest is 56. |
2005 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.