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

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This gives away all 25 answers and the idea behind each one. Sit the paper first if you mean to.

#	Answer	Topic	Key insight
1	D	Exponents, Logarithms & Radicals	Evaluate each parenthesis separately: 2, 6, 12, then combine with the given signs to get 2 - 6 + 12 = 8.
2	C	Linear Equations & Word Problems	The two schools split 2600 students in the ratio 3:1, so Portia's school has three of the four equal parts.
3	D	Bases & Digits	Erasing a trailing zero divides by 10, so the numbers are 10n and n; their sum is 11n and their difference is 9n.
4	D	Sequences & Series	The distances per second form an arithmetic sequence 5, 12, 19, ... with 30 terms; sum it as 30 times the average of the first and last terms.
5	B	Ratios, Percents & Averages	Work with totals, not means: the class sums to 8k, the twelve scores sum to 168, and the remaining k - 12 scores share what is left.
6	A	Linear Equations & Word Problems	Name the half-trail length d; Chantal's three legs take d/4 + d/2 + d/3 = 13d/12 hours, during which Jean covers exactly d.
7	D	Logic Puzzles	Chain the implications: purple -> cannot subtract -> cannot add, while happy -> can add; so a happy snake cannot be purple.
8	E	Fractions & Decimals	The error is the tail 0.00ababab..., which equals ab/9900; so 66 * ab/9900 = 0.5 gives ab = 75.
9	D	Algebraic Manipulation	Expanding, the cross terms -2xy and +2xy cancel and the expression factors as (x^2+1)(y^2+1), which is at least 1.
10	C	Algebraic Manipulation	Multiply by (3 - 2) = 1; repeated difference of squares collapses the whole product to 3^128 - 2^128.
11	E	Bases & Digits	Expand in base b: the difference is 2b^3 - 2b^2 = 2b^2(b-1), divisible by 3 exactly when b or b-1 is a multiple of 3.
12	E	Solid Geometry	Equal volumes with radii 3 and 6 force heights in ratio 4:1; adding equal volumes scales both heights by the same factor, so the rises are also 4:1.
13	C	Solid Geometry	Check Pythagoras on each face at A: 2^2+3^2=13, 2^2+4^2=20, 3^2+4^2=25, so AB, AC, AD are mutually perpendicular.
14	A	Polynomials	Vieta gives root sum 10 and product 16; six positive integers with product 16 are powers of 2, and only 1,1,2,2,2,2 sums to 10.
15	C	Basic Counting	Parabolas meet iff the larger leading coefficient pairs with the larger intercept; swapping B and D exchanges good and bad configurations, so exactly half of 180 work.
16	C	Statistics & Data	There are 20100 numbers, so the median is the 10050th; the first n values occupy n(n+1)/2 slots, and 141*142/2 = 10011 < 10050 <= 10153.
17	D	Similar & Congruent Triangles	CP is perpendicular to BD since CBD is isosceles, so right triangles CPD and ADB are similar with ratio 1:2, giving AB = 86 and BD = 66.
18	E	Functions	Multiplicativity makes f(n) the sum of n's prime factors with multiplicity and f(m/n) = f(m) - f(n); only 25/11 gives 10 - 11 < 0.
19	E	Coordinate Geometry	In the sector \|y\| <= x the equation is x^2 + y^2 = 6x, a circle through the origin; four such arcs bound a 6-by-6 square with four half-disks attached.
20	D	Arrangements with Restrictions	The sequence must zigzag; for the up-down shape, 5 sits at a peak (position 2 or 4), and each choice leaves 8 completions, so 16 total, doubled by reflection.
21	C	Angles & Polygons	Both triangles are equilateral (120-degree angles leave 60-degree corners); one has side AB+BC+FA, the other CD+DE+EF, so the perimeter is their side lengths summed.
22	B	Diophantine Equations	Equating remaining-page total to 19 times remaining-page count gives c(4b + 2c - 37) = 325; the only divisor c of 325 that fits within 25 sheets is 13.
23	D	Conditional Probability & States	Only three states matter: center, edge, corner. From an edge, half the hops reach a corner, a quarter return to the center, a quarter wrap to another edge.
24	D	Coordinate Geometry	Each equation is a pair of parallel lines, and the two directions are perpendicular, so the region is a rectangle whose sides are the distances between the parallel pairs.
25	E	Paths & Grids	The center touches every edge square, so its color's other two chips sit in corners; the leftover squares form paths that the remaining two colors must color alternately.

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