2003 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	Sequences & Series	Pair the k-th even number 2k with the k-th odd number 2k-1; every pair differs by exactly 1, and there are 2003 pairs.
2	B	Linear Equations & Word Problems	Each member's kit costs 2(4 + 9) = 26 dollars, so the head count is 2366 divided by 26.
3	D	Solid Geometry	A box has 8 corners, so 8 cubes of volume 27 are removed from a box of volume 1200.
4	A	Linear Equations & Word Problems	Average speed is total distance over total time: 2 km in 40 minutes, which is 3 km/hr.
5	B	Quadratics	The coefficients 2 + 3 - 5 sum to zero, so x = 1 is a root and the factor (root - 1) kills the product.
6	C	Absolute Value & Inequalities	x heart 0 is \|x\|, which equals x only for nonnegative x; a single negative test value like x = -3 breaks statement (C).
7	B	Triangles: Area & Pythagorean	The longest side must be less than half the perimeter, so it is 3; the other two sides sum to 4, giving only (3,3,1) and (3,2,2).
8	E	Divisibility & Factors	60 = 2^2 * 3 * 5 has 12 divisors, and every integer from 1 to 6 divides 60, so 6 of the 12 qualify.
9	A	Exponents, Logarithms & Radicals	The innermost cube root of x*sqrt(x) = x^{3/2} is exactly sqrt(x), so each layer reproduces sqrt(x) and the whole tower collapses.
10	E	Solid Geometry	Fold the four given squares first; they form a bottom and three walls, and a new square works exactly when it folds onto one of the two missing faces.
11	E	Bases & Digits	Both numbers are 100 times the three-digit number AMC plus 10 or 12, so 200*AMC + 22 = 123422 and AMC = 617.
12	A	Geometric Probability	Inside the 4-by-1 rectangle, x < y only in the small triangle above the line y = x, which has area 1/2 out of 4.
13	A	Systems of Equations	The first number is 4 times the other two combined, so it is 4/5 of the total, namely 16; the remaining 4 splits 7:1.
14	A	Primes	Only e = 3 or 7 lets 10d + e be prime; the four survivors 23, 37, 53, 73 make 7 * 3 * 73 = 1533 the largest product.
15	C	Divisibility & Factors	Count the 50 even numbers, then remove the 16 multiples of 6; 34 of 100 remain.
16	C	Modular Arithmetic	Powers of 3 end in the cycle 3, 9, 7, 1; since 2003 leaves remainder 3 on division by 4, the units digit is the third entry, 7.
17	B	Circles	An equilateral triangle inscribed in a circle of radius r has side rsqrt(3), so 3sqrt(3)r = pir^2 and r = 3*sqrt(3)/pi.
18	B	Quadratics	Substituting y = 1/x turns the equation into y^2 + y + 2003/2004 = 0, whose roots are the reciprocals; Vieta gives their sum as -1 at once.
19	C	Circles	The lune is the small semicircle minus the big circle's segment cut by the small diameter, a chord subtending 60 degrees, so segment = pi/6 - sqrt(3)/4.
20	E	Bases & Digits	Three digits in base b means b^2 <= n < b^3, so n must lie in [121, 728]; that is 608 of the 900 three-digit numbers, about 0.68.
21	D	Distributions & Stars and Bars	An assortment is a nonnegative solution of a + b + c = 6, and stars and bars counts these as C(8,2) = 28.
22	B	Similar & Congruent Triangles	Triangles GCH and GEA are similar with ratio CH:EA = 3:5, and their altitudes from G differ by the rectangle's width 8, so GF = 20.
23	C	Basic Counting	Every toothpick is a side of exactly one upward-pointing small triangle, and there are 1 + 2 + ... + 1002 of those, so multiply that sum by 3.
24	E	Logic Puzzles	Red 5 divides only blue 5 and red 4 only blue 4, so those reds sit at the ends; the rest of the stack is forced card by card.
25	B	Modular Arithmetic	n = 100q + r = 99q + (q + r) and 11 divides 99, so 11 \| q + r means 11 \| n; count five-digit multiples of 11.

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic tags, difficulty ratings and key insights on this page are original to this site. No problem statements are reproduced here — each links to its own page.