2020 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 | E | Linear Equations & Word Problems | Simplify the right side to 1/12 first, then add 3/4 = 9/12 to isolate x. | |
| 2 | C | Ratios, Percents & Averages | An average of 15 over five numbers means the total is 75; remove 3 + 5 + 7 = 15 and halve what remains. | |
| 3 | A | Algebraic Manipulation | Each numerator is the negative of some denominator: (a-3)/(3-a) = -1, and likewise for b and c, so the product is (-1)^3. | |
| 4 | E | Linear Equations & Word Problems | Work per hour: 60 miles earns 30 dollars and burns 2 gallons costing 4 dollars, leaving 26 dollars. | |
| 5 | C | Absolute Value & Inequalities | Split into x^2-12x+34 = 2 (roots 4 and 8) and x^2-12x+34 = -2 (double root 6); sum the distinct roots 4 + 8 + 6. | |
| 6 | B | Basic Counting | Divisible by 5 with an even last digit forces the units digit to be 0; the thousands digit has 4 even choices and each middle digit 5. | |
| 7 | C | Sequences & Series | Whatever the arrangement, the five rows together contain every number once, so the common sum is (total of -10 through 14)/5 = 50/5 = 10. | |
| 8 | B | Sequences & Series | Group the terms in fours: the k-th group is (4k-3)+(4k-2)+(4k-1)-4k = 8k-6, and there are 50 groups. | |
| 9 | B | Divisibility & Factors | With x adults and x children, x/7 + x/11 = N gives 18x = 77N; since gcd(18,77) = 1, N must be a multiple of 18. | |
| 10 | B | Solid Geometry | Seen from above the tower shows exactly a 7-by-7 square of exposed top faces, and likewise from below; only the side faces need summing. | |
| 11 | C | Statistics & Data | With 4040 entries the median averages the 2020th and 2021st; exactly m + 44 entries are at most m (for m near 2000), so m = 1976. | |
| 12 | C | Triangle Centers & Cevians | The centroid cuts each median 2:1, so triangle MPC is a right isosceles triangle with legs 8; its area 32 is one third of the whole triangle. | |
| 13 | B | Conditional Probability & States | Use symmetry to collapse the nine interior points to three unknowns (center is 1/2 by symmetry), then solve a small linear system for the start point. | |
| 14 | D | Algebraic Manipulation | Combine the middle fractions into (x^5 + y^5)/(xy)^2 and build x^5 + y^5 from the power sums using x + y = 4 and xy = -2. | |
| 15 | E | Divisibility & Factors | Factor 12! = 2^10 3^5 5^2 7 11; divisors number 11*6*3*2*2 = 792, square divisors need even exponents: 6*3*2*1*1 = 36, ratio 1/22. | |
| 16 | B | Geometric Probability | Each unit square holds four quarter-discs of radius d that assemble into one full disc, so pi d^2 = 1/2 and d is about 0.4. | |
| 17 | E | Absolute Value & Inequalities | P(n) is negative exactly when an odd number of factors are negative, i.e. n lies strictly between k^2 and (k+1)^2 for odd k; each such gap holds 2k integers. | |
| 18 | C | Number Properties | ad - bc is odd exactly when one product is odd and the other even; a product is odd for 4 of 16 pairs, so 2(4)(12) = 96. | |
| 19 | E | Basic Counting | Routes factor into stages: entry face on the top ring (5), walk along it (9), step down (2), walk the bottom ring (9): 810. | |
| 20 | D | Quadrilaterals & Polygon Areas | Drop the altitude from B to AC: similar triangles put its foot h/2 from E, and the right angle at B gives h^2 = AF*FC, so h = 6. | |
| 21 | C | Bases & Digits | With x = 2^17 the quotient is x^16 - x^15 + ... + 1; each pair x^{2j} - x^{2j-1} is a block of 17 binary ones. | |
| 22 | A | Number Properties | The three floors are equal (sum divisible by 3) unless n divides 999 or 1000; count those divisors, excluding n = 1 which divides both. | |
| 23 | A | Transformations & Symmetry | T has no symmetry, so the composite must be the identity; reflections flip orientation, so there are 0 or 2 of them, and each case gives 6 sequences. | |
| 24 | C | GCD & LCM | The gcds mean 21 | n+120 but not 9, and 60 | n+63 but not 8; solve n = 237 mod 420, then test 1077, 1497, 1917. | |
| 25 | A | Basic Probability | Keeping one die a and rerolling two wins with probability (6-a)/36; that is optimal only when every pair sums to at least 7 and a is at most 3. |
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