2021 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 123 KB — a separate file from the paper, so one can be handed over without the other.

#	Answer	Topic	Key insight
1	D	Exponents, Logarithms & Radicals	Work out each group in parentheses first. They give 2, 6 and 12, and then 2 - 6 + 12 = 8.
2	C	Linear Equations & Word Problems	Lara's school is one share and Portia's is three shares, so the 2600 students make four equal shares.
3	D	Bases & Digits	Erasing the final 0 of a number divides it by 10. So the two numbers are 10 times n and n, which add up to 11 times n.
4	D	Sequences & Series	The distances 5, 12, 19, ... grow by 7 each second. Find the distance in second 30, then add all thirty by pairing the first with the last.
5	B	Ratios, Percents & Averages	Turn every mean into a total. The class total is 8k, the twelve scores total 168, and the other students share what is left.
6	A	Linear Equations & Word Problems	Chantal and Jean walk for the same amount of time. Pick an easy halfway distance, such as 12 miles, and add up Chantal's three walking times.
7	D	Logic Puzzles	Follow the chain: a purple snake can't subtract, so it can't add. Every happy snake can add, so no happy snake can be purple.
8	E	Fractions & Decimals	The student only lost the tail 0.00ababab... Multiplying that lost part by 66 must give exactly 0.5.
9	D	Algebraic Manipulation	Multiply out both squares. The middle terms cancel, leaving three squares plus 1, so the value is never below 1, and x = y = 0 gives exactly 1.
10	C	Algebraic Manipulation	Multiply by (3 - 2), which equals 1. Each factor then combines with the one before by difference of squares, and the whole product collapses to 3^128 - 2^128.
11	E	Bases & Digits	Write both numbers using powers of b. Their difference is 2 × b × b × (b - 1), a multiple of 3 whenever b or b - 1 is.
12	E	Solid Geometry	Equal volumes make the narrow cone's liquid 4 times as deep. Equal marbles stretch both depths by the same factor, so the rises stay 4 to 1.
13	C	Solid Geometry	Check the three faces at corner A with the Pythagorean theorem. All three angles at A are right angles, so A is like the corner of a box.
14	A	Polynomials	The roots add to 10 and multiply to 16. Positive whole numbers with product 16 are powers of 2; only 2, 2, 2, 2, 1, 1 adds to 10.
15	C	Basic Counting	Curves meet only if the steeper parabola has the smaller constant. Swapping the two constants turns each good choice into a bad one, so half of all choices work.
16	C	Statistics & Data	The list has 20100 numbers, so the middle is at position 10050. The numbers 1 through n fill the first n(n+1)/2 spots; find which block holds that position.
17	D	Similar & Congruent Triangles	Since CB = CD, the segment from C to the midpoint P of BD is perpendicular to BD. That makes triangle CPD a half-size copy of triangle ADB.
18	E	Functions	The rule turns products into sums, so f of a whole number is the sum of its prime factors, repeats included, and f of a fraction is top minus bottom.
19	E	Coordinate Geometry	In the right-hand quarter between y = x and y = -x the equation becomes a circle; four copies give a 6 by 6 square with four half-disks attached.
20	D	Arrangements with Restrictions	The terms must go up, down, up, down, or the reverse. In the up-down pattern 5 must sit in position 2 or 4, and each spot gives 8 ways.
21	C	Angles & Polygons	Every angle is 120 degrees, so extending every other side makes an equilateral triangle. Each big triangle's side is three hexagon sides, and the two triangles together use all six.
22	B	Diophantine Equations	Write the total of the pages left two ways, then set them equal. The equation becomes (sheets taken) times (a whole number) = 325, so try factors of 325.
23	D	Conditional Probability & States	Only three kinds of square matter: center, edge, corner. From an edge, half the hops reach a corner, a quarter return to center, a quarter wrap to an edge.
24	D	Coordinate Geometry	Each equation is two parallel lines, and the pairs meet at right angles, so the region is a rectangle. The formula works for every a, so try a = 2.
25	E	Paths & Grids	The center touches all four edge squares, so its color's other two chips must go in corners. The empty squares then form paths the other two colors alternate along.

2021 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.