2001 AMC 10

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	E	Statistics & Data	The list is already sorted, so the median is the fifth entry n+6; the mean is n plus the average of the offsets, 63/9 = 7.
2	C	Linear Equations & Word Problems	The reciprocal 1/x times the additive inverse -x is always -1, so the number is simply -1 + 2 = 1.
3	E	Linear Equations & Word Problems	Adding 3 to each of two numbers adds 6 to their sum, and doubling each number doubles the sum, giving 2(S + 6) = 2S + 12.
4	E	Circles	A line meets a circle at most twice, so three sides give at most six points, and a triangle slightly larger than the circle achieves all six.
5	D	Transformations & Symmetry	Check each pentomino for a mirror axis, including diagonal ones: I, T, U, V, W and X have one; F, L, N, P, Y and Z do not.
6	E	Bases & Digits	Write N = 10a + b; the condition becomes 10a + b = ab + a + b, and cancelling b and dividing by a leaves b = 9.
7	C	Fractions & Decimals	Moving the decimal point four places right multiplies by 10^4, so 10^4 x = 4/x gives x^2 = 4 * 10^(-4) and x = 2 * 10^(-2).
8	B	GCD & LCM	They coincide again after a number of days divisible by 3, 4, 6 and 7, so the answer is lcm(3,4,6,7) = 84.
9	B	Ratios, Percents & Averages	Split the tax as p% of all income plus an extra 2% on the amount over 28000; then the p% terms cancel and 2%(I - 28000) = 0.25% I.
10	D	Systems of Equations	Dividing two of the equations cancels a variable: xz/xy = 2 gives z = 2y, and then yz = 2y^2 = 72 pins down y = 6.
11	C	Sequences & Series	The nth ring is a (2n+1)-square minus a (2n-1)-square, and (2n+1)^2 - (2n-1)^2 = 8n, so the 100th ring has 800 squares.
12	D	Divisibility & Factors	Three consecutive integers always supply a factor of 2 and a factor of 3, but not necessarily 4; test one example like 567.
13	E	Logic Puzzles	All ten digits are used, so ABC is whatever the odd run GHIJ and even run DEF leave behind; only 9753 and 642 leave digits summing to 9.
14	A	Diophantine Equations	Doubling the revenue equation gives p(f + 140) = 4002 = 2323*29, and the only divisor between 140 and 280 is 174, so f = 34 and p = 23.
15	C	Quadrilaterals & Polygon Areas	The crosswalk is a parallelogram: its area is 15 times 40 using the curbs as bases and 50 times d using the stripes, so d = 12.
16	D	Statistics & Data	Write the least and greatest as m - 10 and m + 15 in terms of the mean m; with median 5 the sum 2m + 10 must equal 3m.
17	C	Solid Geometry	The sector's radius becomes the slant height and its arc becomes the base circumference: (252/360)(20 pi) = 14 pi, so the base radius is 7.
18	D	Quadrilaterals & Polygon Areas	One repeating 3 by 3 block has area 9, of which the four corner unit squares take 4, so the four pentagons cover 5/9, about 55.6 percent.
19	D	Distributions & Stars and Bars	Choosing 4 donuts of 3 types is the number of nonnegative solutions to g + c + p = 4, which stars and bars counts as C(6,2).
20	B	Angles & Polygons	One side of the square holds two triangle legs plus one octagon side s; each leg is s/sqrt(2), so s(1 + sqrt 2) = 2000.
21	B	Solid Geometry	Slice through the axis: the cone becomes a triangle, the cylinder a 2r-by-2r square, and the small triangle above it is similar to the whole.
22	D	Systems of Equations	Compare lines that share an unknown: left column versus main diagonal gives x = 22, top row versus left column gives w = 19 and the magic sum 66.
23	D	Basic Probability	Pretend all five chips are drawn; the last chip actually drawn is white exactly when the very last chip of the full sequence is red, which has probability 3/5.
24	B	Triangles: Area & Pythagorean	Drop the perpendicular from B to CD: a right triangle with legs 7 and CD - AB and hypotenuse AB + CD, so 4(AB)(CD) = 49.
25	B	Inclusion-Exclusion	Count multiples of 3 or 4 by inclusion-exclusion, then remove only those that are also multiples of 5 by a second inclusion-exclusion with 15, 20 and 60.

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