2017 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.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Sequences & Series | Evaluate from the inside out; every layer doubles and adds 1, so the values run 3, 7, 15, 31, 63, 127, each one less than a power of 2. | |
| 2 | D | Linear Equations & Word Problems | The 5-boxes are the best deal at 60 cents per popsicle; buy as many as possible (two), then spend the leftover $2 on a 3-box. | |
| 3 | B | Quadrilaterals & Polygon Areas | Walkway area is the whole garden minus the six beds: the garden is 15 by 10 once you add the 1-foot walkways between and around the beds. | |
| 4 | B | Games & Processes | Each 30-second cycle nets one toy, but the box reaches 30 the instant Mom drops in her third batch, before Mia can react, so 27 cycles plus one more deposit. | |
| 5 | C | Algebraic Manipulation | The sum of reciprocals 1/x + 1/y equals (x+y)/(xy), and the given relation x+y = 4xy makes that quotient exactly 4. | |
| 6 | B | Logic Puzzles | The promise is 'all right implies an A'; the only statement forced by it is the contrapositive: no A implies at least one question wrong. | |
| 7 | A | Triangles: Area & Pythagorean | With side 1, Jerry walks 2 and Silvia walks the diagonal sqrt(2) of about 1.41, so she saves about 0.59 out of 2, roughly 30%. | |
| 8 | B | Basic Counting | A handshake happens exactly when at least one of the pair is a stranger: 10 times 20 mixed pairs plus C(10,2) stranger-stranger pairs. | |
| 9 | C | Linear Equations & Word Problems | Compute each leg as distance over speed; Penny rides the loop backwards, so her uphill leg is the 15 km stretch and her downhill leg is the 10 km one. | |
| 10 | B | Triangles: Area & Pythagorean | A quadrilateral exists exactly when the longest side is shorter than the other three combined, giving 5 < d < 25, then remove the used lengths 7 and 15. | |
| 11 | D | Solid Geometry | Points within 3 of a segment form a cylinder of radius 3 capped by two hemispheres, i.e. a cylinder plus one full sphere of radius 3. | |
| 12 | E | Coordinate Geometry | Each pair of equal quantities gives a line, and the 'third is no greater' condition trims each line to a ray; all three rays start at (1, 7). | |
| 13 | D | Modular Arithmetic | Fibonacci mod 3 repeats with period 8, so any eight consecutive terms are a full cycle and sum to 0+1+1+2+0+2+2+1 = 9. | |
| 14 | D | Ratios, Percents & Averages | Turn the percent statements into 5m = A - s and 20s = A - m; solving gives m = 19A/99 and s = 4A/99, total 23A/99. | |
| 15 | C | Geometric Probability | Half the time Laurent's number exceeds 2017 and automatically wins; the other half both numbers are uniform on the same interval, and he wins half of that. | |
| 16 | B | Divisibility & Factors | Horse k is at the start at time T exactly when k divides T, so T is the least number with five divisors among 1 to 10: T = 12. | |
| 17 | D | Coordinate Geometry | Only 12 lattice points lie on the circle; the longest irrational chord is sqrt(98) = 7 sqrt(2) and the shortest is sqrt(2), so the ratio is 7. | |
| 18 | D | Conditional Probability & States | If both first tosses fail (probability 2/5), the game restarts from the same state, so p = 1/3 + (2/5) p, giving p = 5/9. | |
| 19 | C | Arrangements with Restrictions | Alice's neighbors must be Derek and Eric; place Alice first, and only when she sits at an end can Derek and Eric become adjacent, which costs a few arrangements. | |
| 20 | D | Bases & Digits | Adding 1 to a number ending in k nines changes the digit sum by 1 - 9k, so S(n+1) is 1275 minus a multiple of 9; only 1239 fits. | |
| 21 | D | Similar & Congruent Triangles | Each square cuts off triangles similar to the 3-4-5; the corner square gives x = 12/7, and the hypotenuse splits as 4y/3 + y + 3y/4 = 5. | |
| 22 | E | Circles | Radii to the tangent points make angle BOC = 120 degrees, so the triangle's overlap with the circle is a 120-degree segment; compare it to the triangle's area. | |
| 23 | B | Basic Counting | Count all C(25,3) triples and subtract collinear ones, organized by the line's direction: rows, columns, both diagonal directions, and slopes of plus or minus 1/2 and 2. | |
| 24 | C | Polynomials | Three shared distinct roots force f(x) = g(x)(x - r); the x coefficient gives r = -90, then a = -89 and f(1) = g(1)(1 - r). | |
| 25 | A | Basic Counting | By the 11-test a digit triple works iff one digit equals the sum of the other two, or that sum minus 11; count triples by type, then arrangements. |
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