2002 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.
The same key as a PDF 126 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | E | Exponents, Logarithms & Radicals | Six is 2 times 3, so write the bottom as powers of 2 and 3. Then cancel twos against twos and threes against threes. | |
| 2 | C | Functions | The rule D just says: multiply the three numbers, add the three numbers, and divide the product by the sum. | |
| 3 | A | Bases & Digits | Every number made of nines is 9 times a number made of ones. So the mean is 1 + 11 + 111 + ... + 111111111, which is 123456789. | |
| 4 | D | Algebraic Manipulation | The first two parts both contain (3x - 2). Pull it out and the rest cancels, leaving 3x - 1, which is 11 when x = 4. | |
| 5 | E | Circles | The two small circles sit side by side across the big circle, so the big circle's diameter is 6 + 4 = 10 and its radius is 5. | |
| 6 | B | Primes | The expression factors as (n - 1)(n - 2), two whole numbers next to each other, so it is always even. The only even prime is 2. | |
| 7 | E | Fractions & Decimals | The known fractions add to 41/42, just under 1. Since 1/n is at most 1, the sum stays below 2, so it must equal 1 and n = 42. | |
| 8 | D | Clocks, Calendars & Time | A 31-day month has three weekdays that appear five times: those of the 1st, 2nd and 3rd. Try each possible start for July and see which August day always appears. | |
| 9 | D | Basic Counting | Count the words that come before USAMO in blocks: first all words starting with A, M, O or S, then the words starting UA, UM or UO. USAMO is next. | |
| 10 | C | Quadratics | With solutions a and b, the quadratic is (x-a)(x-b). Matching terms gives a+b = -a and ab = b, so a = 1 and b = -2. | |
| 11 | B | Algebraic Manipulation | Call the middle number n. The sum is 3n and the product is (n-1)n(n+1), so the condition becomes (n-1)(n+1) = 24, giving n = 5. | |
| 12 | E | Linear Equations & Word Problems | Cross-multiply. The x squared terms cancel, leaving (k-5)x = 2k-6. At k = 5 the left side is always 0 but the right side is 4. | |
| 13 | D | Algebraic Manipulation | Write it as (8x-12)y + (2x-3). For it to be zero for every y, the part multiplying y and the leftover part must both be zero. | |
| 14 | B | Exponents, Logarithms & Radicals | N is 5 to the 64th times 2 to the 75th. Pair each 5 with a 2 to make 64 tens; 2048 is left over. | |
| 15 | E | Primes | A-B and A+B are both odd primes, so B must be the even prime 2. Then A-2, A, A+2 are all prime, which only 3, 5, 7 manage. | |
| 16 | D | Diophantine Equations | A square is never negative, so n is from 0 to 19 and the fraction is at most 19. Only the squares 0, 1, 4, 9 give whole-number n. | |
| 17 | C | Quadrilaterals & Polygon Areas | See the octagon as a square with corners cut off. DG spans that square, length 2 + 2 root 2, and A is 2 + root 2 below DG. | |
| 18 | D | Circles | Two circles cross in at most 2 points. Four circles make 6 pairs, so at most 12 points, and slightly shifted copies of one circle reach 12. | |
| 19 | C | Sequences & Series | Each term in the second block is 100 steps after its partner in the first, so it is 100d bigger. The sums differ by 10000d = 100. | |
| 20 | B | Algebraic Manipulation | Move b to the right, square both equations and add. The mixed terms cancel, leaving 65(a^2 + c^2) = 65(1 + b^2). | |
| 21 | B | Ratios, Percents & Averages | Only comparisons are given, so pick easy numbers: lawns 6, 3, 2 and mowing speeds 3, 2, 1. Then times are 2, 1.5 and 2, so Beth finishes first. | |
| 22 | B | Triangles: Area & Pythagorean | Call OM = x and ON = y. The two Pythagorean equations are mirror images; adding them gives x^2 + y^2 = 169, and XY is twice 13. | |
| 23 | D | Sequences & Series | Put n = 1: each term is the previous one plus the next counting number. That gives 1, 3, 6, 10, ..., the triangular numbers, so a_12 = 78. | |
| 24 | D | Circles | Ten feet up is halfway from the bottom to the center, so bottom, center and rider form an equilateral triangle: a 60 degree turn, one sixth of a minute. | |
| 25 | A | Linear Equations & Word Problems | Call the count n and the mean m, so the total is mn. Each mean change gives one equation: m + 2n = 13 and 2m + n = 14. |
2002 AMC 10B. 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.