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

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

#	Answer	Topic	Key insight
1	D	Sequences & Series	Match the first even number with the first odd number, the second with the second, and so on. Each even number is exactly 1 more than its odd partner.
2	B	Linear Equations & Word Problems	Work out what one member spends: two pairs of socks and two shirts, which is 26 dollars. Then divide the total bill by 26.
3	D	Solid Geometry	A box has 8 corners, not 4. So 8 small cubes of volume 27 are cut out of a box of volume 1200.
4	A	Linear Equations & Word Problems	Average speed is total distance divided by total time: 2 km in 40 minutes, which is 3 km per hour.
5	B	Quadratics	Putting x = 1 into the equation gives zero, so 1 is a solution. One factor is then 1 minus 1, and the whole product is zero.
6	C	Absolute Value & Inequalities	x heart 0 is the absolute value of x, which is never negative. So for a negative x, like -3, it cannot equal x.
7	B	Triangles: Area & Pythagorean	The longest side must be shorter than the other two together, so it is less than half of 7. That makes it 3, and the others add to 4.
8	E	Divisibility & Factors	60 has 12 factors, and every whole number from 1 to 6 divides 60, so 6 of the 12 factors are less than 7.
9	A	Exponents, Logarithms & Radicals	Start at the innermost part: the cube root of x times the square root of x is just the square root of x. Every layer then repeats the same step.
10	E	Solid Geometry	Fold the four solid squares first. They make a bottom and three walls, so a fifth square works exactly when it folds onto one of the two faces still missing.
11	E	Bases & Digits	Both numbers start with the same three digits AMC. Written as place values, the sum is twice AMC hundreds, plus 10 plus 12, which gives AMC = 617.
12	A	Geometric Probability	Probability equals area here. The points with x less than y fill only a small triangle of area one half, inside a rectangle of area 4.
13	A	Systems of Equations	Treat the second and third numbers together as one amount. The first is 4 times that amount, so the total 20 is 5 of those amounts.
14	A	Primes	d and e must be among 2, 3, 5, 7, and a two-digit prime cannot end in 2 or 5. That leaves only a few cases to list and compare.
15	C	Divisibility & Factors	Count the 50 even numbers, then take away the even numbers that are also multiples of 3, which are the 16 multiples of 6. That leaves 34.
16	C	Modular Arithmetic	Only the last digit matters, so look at powers of 3. Their last digits repeat 3, 9, 7, 1, and 2003 is 3 past a multiple of 4.
17	B	Circles	In an equilateral triangle drawn inside a circle, each side equals the radius times the square root of 3. Set 3 times that side equal to pi times radius squared.
18	B	Quadratics	Replace x by 1 over y. The new equation, y squared plus y plus 2003/2004 = 0, has the reciprocals as its solutions, and they add to -1.
19	C	Circles	The lune is the small half-circle minus the piece of the big circle above its diameter. That piece is a 60 degree slice minus an equilateral triangle of side 1.
20	E	Bases & Digits	Three digits in base 9 means 81 to 728; in base 11, 121 to 1330. Both hold from 121 to 728: 608 of the 900 numbers.
21	D	Distributions & Stars and Bars	Cookies of the same kind are identical, so an assortment is just how many of each kind. Count the ways three whole numbers can add to 6.
22	B	Similar & Congruent Triangles	G sits above two parallel sides. The triangles from G to each side have the same shape, with bases 3 and 5, and heights differing by 8.
23	C	Basic Counting	Every toothpick is a side of exactly one upward-pointing small triangle. Count those, 1 + 2 + ... + 1002, and multiply by 3.
24	E	Logic Puzzles	Red 5 fits beside only blue 5, and red 4 beside only blue 4. So those reds go at the two ends, and every other card is forced.
25	B	Modular Arithmetic	Write n as 99 times q plus the sum q + r. Since 11 divides 99, q + r is a multiple of 11 exactly when n is.

2003 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.