2021 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.

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

#	Answer	Topic	Key insight
1	D	Absolute Value & Inequalities	3π is about 9.42, so x can be any whole number from −9 up to 9. That is 9 negatives, 9 positives, and zero.
2	D	Exponents, Logarithms & Radicals	The square root of a square is never negative. Since 2√3 (about 3.46) is bigger than 3, the first bracket is negative and becomes 2√3 − 3.
3	C	Ratios, Percents & Averages	A quarter of the juniors equals a tenth of the seniors, so for every 2 juniors there are 5 seniors. Split 28 into 2 and 5 parts: 8 juniors.
4	B	Linear Equations & Word Problems	The 23 blue–blue pairs use 46 blue students. The other 11 blue students each sit with a yellow, which leaves 64 yellows to pair with each other: 32 pairs.
5	B	Divisibility & Factors	Using single digits, 30 can only be 5 × 6. Since 6 is now taken, 24 has to be 3 × 8, not 4 × 6.
6	C	Statistics & Data	Only the 3 to 4 ratio matters, so pretend there are 3 morning and 4 afternoon students. Their scores total 532 over 7 students, which is 76.
7	D	Circles	Circles on one side sit inside each other, so a side counts only its biggest circle minus its second biggest. Put 7 alone: 49 + 16 = 65.
8	A	Paths & Grids	Odd squares sit at the top-right corners of the rings. 169 ends the 13 by 13 ring in row 2, so row 2 holds 157 to 170, plus 210.
9	D	Coordinate Geometry	Work backward and undo the last move first: reflecting (−6, 3) over y = −x gives (−3, 6). Then turn that point a quarter turn clockwise around (1, 5).
10	A	Solid Geometry	Pouring keeps the amount of water the same. Set the cone's volume, (1/3)π · 12² · 18, equal to π · 24² · h, so h = 1.5.
11	D	Diophantine Equations	The inside pieces must be half of all pieces: 2(m − 2)(n − 2) = mn. Only two pans work, 5 by 12 and 6 by 8.
12	C	Divisibility & Factors	Every even divisor is an odd divisor times 2, 4, or 8. So the even divisors add up to 2 + 4 + 8 = 14 times the odd ones.
13	B	Bases & Digits	Both base-n numbers start 3n² + 2n. Subtracting the two equations cancels that part and gives d = 2, and then 3n² + 2n = 261 gives n = 9.
14	B	Circles	Equal chords are equally far from the center, so it sits halfway between the 38-chords. Then 19² + (d/2)² = 17² + (3d/2)².
15	B	Algebraic Manipulation	Square twice to get x⁴ + 1/x⁴ = 7. Multiply that by x⁷ and you get x¹¹ + x³ = 7x⁷, so the expression is 0.
16	C	Divisibility & Factors	Uphill numbers cannot end in 0, so a multiple of 5 ends in 5. Then choose digits from 1 to 4 so the digit sum is a multiple of 3.
17	C	Logic Puzzles	Start with the smallest score. 4 can only be 1 + 3. Then 7 must be 2 + 5, and 11 must be 4 + 7.
18	C	Basic Probability	Rolls that repeat a number already seen change nothing, so skip them. Then the first three different numbers to appear must be 2, 4, 6: (1/2)(2/5)(1/4) = 1/20.
19	D	Statistics & Data	Removing the greatest or the least leaves n − 1 numbers either way. Those two totals differ by 72 and the averages by 8, so n − 1 = 9.
20	D	Quadrilaterals & Polygon Areas	Triangles ABF and BCF are equilateral, so the diagonal AC is 2√3, and the same holds for AD. Cut the pentagon into triangles ABC, ACD, ADE.
21	A	Transformations & Symmetry	Folding keeps lengths, so the small right triangle at D has sides 1/3, 4/9, 5/9. Triangle AEC' has the same shape, 3/2 times as big, so its perimeter is 2.
22	D	Inclusion-Exclusion	Ang's order can be taken as fixed. A box is all one color when Ben and Jasmin both match Ang there. Count with add, subtract, add: 2556 of 14400.
23	C	Geometric Probability	Follow the coin's center. It lands in a 7 by 7 square, and the coin touches black exactly when the center is within 1/2 of a black shape.
24	B	Games & Processes	If the walls come in identical pairs, the second player wins by copying each move on the twin wall. In every wrong choice, Arjun can make such pairs at once.
25	E	Coordinate Geometry	The area under the line is 450m, so try slope 2/3: exactly 300 points. The count stays 300 until the line reaches the point (28, 19).

2021 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.