2005 AMC 10B
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 | A | Ratios, Percents & Averages | Convert both prices to a per-bar rate (40 cents to buy, 50 cents to sell) so the profit is 10 cents per bar times 1000 bars. | |
| 2 | D | Ratios, Percents & Averages | Translate x percent of x into (x/100) times x, so x squared equals 400 and x is 20. | |
| 3 | D | Ratios, Percents & Averages | Each day leaves two thirds of what was there, so after two days two thirds of two thirds, or four ninths, remains. | |
| 4 | D | Functions | Squaring erases the signs, so both inner operations give sqrt(25+144) = 13, and 13 diamond 13 is sqrt(2 * 169) = 13 sqrt 2. | |
| 5 | C | Ratios, Percents & Averages | If one fifth of the money buys one third of the CDs, all the CDs cost three times that, three fifths, leaving two fifths. | |
| 6 | B | Ratios, Percents & Averages | The goal is 40 A's out of 50; she has 22, so 18 of the last 20 quizzes must be A's, leaving at most 2 non-A's. | |
| 7 | B | Circles | Each inscription step halves the area: a square inside a circle inside a square has half the area of the outer square, and the same for circles. | |
| 8 | A | Circles | The four quarter circles on one tile assemble into one full circle of radius 1/2, so each tile has 1 - pi/4 shaded, and there are 80 tiles. | |
| 9 | D | Basic Probability | The sum is odd exactly when the two dice show different parities; the first die is odd with probability 2/3 and the second with probability 1/3. | |
| 10 | A | Triangles: Area & Pythagorean | The altitude from C hits the midpoint of AB; its length squared is 48, so the foot is 4 from D by Pythagoras and BD = 4 - 1. | |
| 11 | E | Sequences & Series | After the first term the sequence cycles 133, 55, 250 with period 3, and term 2005 sits in the same slot as term 4. | |
| 12 | E | Basic Probability | A prime product forces eleven 1's and a single die showing 2, 3 or 5; count 12 positions times 3 primes over 6^12. | |
| 13 | C | Inclusion-Exclusion | Multiples of 12 are counted once in the 3-list and once in the 4-list, so to exclude them entirely subtract the 12-count twice. | |
| 14 | C | Triangles: Area & Pythagorean | Use CD = 2 as the base; the height from M to line BD is half the height of the equilateral triangle, sqrt3/2, since M is a midpoint. | |
| 15 | D | Basic Probability | A sum of at least 20 needs a twenty or both tens: 28 - C(6,2) = 13 pairs contain a twenty, plus 1 pair of tens, out of 28. | |
| 16 | D | Quadratics | Doubling both roots doubles the sum and quadruples the product, so Vieta gives m = 2p and n = 4m, hence n = 8p. | |
| 17 | B | Exponents, Logarithms & Radicals | Substitute each equation into the next: 8 = 7^d = 6^{cd} = 5^{bcd} = 4^{abcd}, so 2^3 = 2^{2abcd} and abcd = 3/2. | |
| 18 | D | Basic Counting | An increasing string of 7 distinct digits is determined by which 7 of the 8 allowed digits are used, i.e. which single digit is left out. | |
| 19 | B | Statistics & Data | The mean is the percent-weighted average, 86; the median is the score where the cumulative percent first passes 50, which is 85. | |
| 20 | C | Basic Counting | By symmetry each place holds each digit equally often, so the average number has the average digit 4.8 in every place: 4.8 times 11111. | |
| 21 | A | Basic Probability | Both probabilities share the denominator C(40,4), so q/p is just a ratio of favorable counts: 45 pairs of numbers times 6 times 6, over 10. | |
| 22 | C | Divisibility & Factors | n!/(1+...+n) = 2(n-1)!/(n+1), which fails to be an integer exactly when n+1 is an odd prime; eight such n up to 24, so 24 - 8 = 16. | |
| 23 | C | Quadrilaterals & Polygon Areas | EF is the midsegment, so EF = (AB + DC)/2, and the two half-height trapezoids have areas proportional to AB + EF and EF + DC. | |
| 24 | E | Diophantine Equations | x^2 - y^2 = 99(a+b)(a-b); the lone factor 11 forces a + b = 11, then a - b must be a perfect square, giving 65 and 56. | |
| 25 | C | Sets, Estimation & Miscellaneous | Numbers 1 to 24 are never in a bad pair; 25 to 100 form 38 pairs summing to 125, each giving at most one element: 24 + 38 = 62. |
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.