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.

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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 125 KB — a separate file from the paper, so one can be handed over without the other.

#	Answer	Topic	Key insight
1	E	Linear Equations & Word Problems	Work out the right side first (it is 1/12), then add 3/4 to both sides. Using twelfths for every fraction keeps the arithmetic small.
2	C	Ratios, Percents & Averages	Five numbers averaging 15 add up to 75. Take away 3 + 5 + 7 = 15 to get a + b = 60, then halve it.
3	A	Algebraic Manipulation	Each top is a bottom with its sign flipped, like a - 3 and 3 - a. Matched up, each fraction is -1, so the product is -1.
4	E	Linear Equations & Word Problems	The answer is per hour, so look at one hour. She drives 60 miles, earns 30 dollars, uses 2 gallons costing 4 dollars, and keeps 26 dollars.
5	C	Absolute Value & Inequalities	The inside must equal 2 or -2. The first case gives x = 4 and x = 8; the second gives only x = 6. Add the three different numbers.
6	B	Basic Counting	The last digit must be 0 (even and divisible by 5). The first digit has 4 choices, since it cannot be 0, and each middle digit has 5.
7	C	Sequences & Series	The five rows together use every number exactly once. So five times the row sum equals the total of all 25 numbers, which is 50. The row sum is 10.
8	B	Sequences & Series	Cut the sum into 50 groups of four. The groups add to 2, 10, 18, and so on, going up by 8 each time. Add those 50 numbers.
9	B	Divisibility & Factors	Cut a bench into 77 pieces: an adult uses 11 and a child 7, so each adult-child pair uses 18. So 77 times N must be a multiple of 18.
10	B	Solid Geometry	Side faces are never covered. From above you see exactly a 7 by 7 square of top faces, and from below another 7 by 7 square.
11	C	Statistics & Data	With 4040 numbers the median is halfway between the 2020th and 2021st. The squares 1, 4, ..., 1936 (44 of them) squeeze in below, pushing the middle down by 44.
12	C	Triangle Centers & Cevians	The medians cross two thirds of the way along each, so triangle MPC has legs 8 and 8 and area 32. It is one third of the whole.
13	B	Conditional Probability & States	The points (1,1), (1,3) and the center sit on diagonals of the square, so from them both kinds of side are equally likely. Look one jump ahead.
14	D	Algebraic Manipulation	Combine the fractions over (xy) squared, which is 4. Then build the sum of fifth powers from the sum 4 and product -2, never finding x or y.
15	E	Divisibility & Factors	Write 12! as prime powers. A divisor chooses a power of each prime, and it is a square exactly when every chosen power is even.
16	B	Geometric Probability	All unit squares look alike. The points within d of the four corners form four quarter-circles, one full circle of area pi d squared. Set that equal to 1/2.
17	E	Absolute Value & Inequalities	P(n) is zero at the 100 squares. Between squares its sign flips each time, so it is negative in every other gap, starting with the gap from 1 to 4.
18	C	Number Properties	The difference is odd when exactly one of the two products is odd. A product of two numbers from 0 to 3 is odd for only 4 of 16 pairs.
19	E	Basic Counting	Every route goes: top face, along the top ring, down, along the bottom ring, bottom face. Each ring walk has 9 options, so 5 times 9 times 2 times 9.
20	D	Quadrilaterals & Polygon Areas	Diagonal AC splits the shape into two triangles on one base. ACD is easy. For ABC, find the height from B using similar triangles at E and the right angle.
21	C	Bases & Digits	Let x be 2 to the 17th. The fraction becomes x^16 - x^15 + ... + 1, and each pair like x^16 - x^15 is seventeen ones in binary.
22	A	Number Properties	The three floors are equal, so their sum is a multiple of 3, unless n divides 999 or 1000. Count those divisors, leaving out n = 1.
23	A	Transformations & Symmetry	T has no symmetry, so every point must return home. A reflection makes a mirror image, so there must be 0 or 2 reflections. Each case gives 6.
24	C	GCD & LCM	Each gcd says a number divides and a bigger power does not. The dividing parts give n = 237 plus multiples of 420; test 1077, 1497, 1917.
25	A	Basic Probability	Keeping one die and rerolling two is best only when every pair of dice adds to 7 or more and the smallest die is 1, 2 or 3.

2020 AMC 10A. 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.