2003 AMC 10B
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.
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 125 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Algebraic Manipulation | Each number on top is 2 times a counting number and the number right under it is 3 times the same counting number, so the fraction is 2/3. | |
| 2 | D | Linear Equations & Word Problems | Divide by the 14 days first: one green plus one pink costs 39 dollars. Two prices that add to 39 and differ by 1 are 20 and 19. | |
| 3 | B | Sequences & Series | The first 8 odd numbers add to 64, so the five even numbers add to 60. Their middle one is the average, 12, so the smallest is 8. | |
| 4 | A | Quadrilaterals & Polygon Areas | Find the five areas from the figure (21, 20, 15, 6, 4). To spend the least, put the cheapest flower in the biggest region and the priciest in the smallest. | |
| 5 | C | Linear Equations & Word Problems | Because of the 4-inch overlap, each pass mows only 24 inches = 2 feet of new grass. Walking 5000 feet an hour then clears 10000 square feet an hour. | |
| 6 | D | Triangles: Area & Pythagorean | A screen 4 wide and 3 tall has a diagonal of 5, so the width is always 4/5 of the diagonal: 4/5 of 27 is 21.6. | |
| 7 | B | Number Properties | The rounded-down square root stays the same between perfect squares: it is 1 three times, 2 five times, 3 seven times, then 4 once at 16. | |
| 8 | B | Sequences & Series | From the 2nd term to the 4th you multiply by the ratio twice, so the ratio squared is 3. The ratio may be negative. | |
| 9 | B | Exponents, Logarithms & Radicals | Write every number as a power of 5. Then the exponents must match: -4 = -12/x, so x = 3. | |
| 10 | C | Basic Counting | Old plates: 26 times 10 four times. New plates: 26 three times times 10 three times. Divide new by old and cancel: 26 squared over 10. | |
| 11 | A | Coordinate Geometry | Both lines drop 15 units to reach the x-axis: slope 3 needs 5 steps left, slope 5 needs 3 steps left. The crossings are 5 and 7. | |
| 12 | C | Linear Equations & Word Problems | Treat Betty and Clare as one team whose money doubles. If Al starts with a, the team has 1000 - a, and the total 1500 gives a = 400. | |
| 13 | E | Bases & Digits | A two-digit number's digits add to at most 18, and only 3 and 12 in that range have digit sum 3. Count both cases. | |
| 14 | D | Exponents, Logarithms & Radicals | The exponent b must go evenly into both 8 and 2, so b is at most 2. Then a = 405 and a + b = 407. | |
| 15 | E | Games & Processes | Every match knocks out exactly one player, and 99 players must be knocked out to leave one winner. So there are 99 matches, and 99 = 9 times 11. | |
| 16 | E | Basic Counting | There are 6 times m times m dinners, which must reach 365. Seven main courses give 294, too few; eight give 384. | |
| 17 | B | Solid Geometry | The melted ice cream is 3/4 of the sphere's volume, which simplifies to pi times r cubed. Setting that equal to the cone's volume gives height = 3 times radius. | |
| 18 | D | Divisibility & Factors | Five odd numbers in a row always include a multiple of 3 and a multiple of 5, but not always a multiple of 11. | |
| 19 | E | Circles | Neighboring small semicircles cross at a point 1 away from both centers, forming equilateral triangles. The white part is two triangles plus slices totaling 5/6 of a circle. | |
| 20 | D | Similar & Congruent Triangles | FG = 2 is parallel to AB = 5, so the top triangle is a 2/5 copy of triangle AEB. Their heights differ by 3, so the height is 5. | |
| 21 | C | Conditional Probability & States | All beads end red exactly when two of the three draws are green. Remember the bag holds fewer greens after each green draw. | |
| 22 | B | Clocks, Calendars & Time | A full day has 180 chimes and the rest of February 26 has 91. The other 1912 chimes are 10 full days plus 112: March 9. | |
| 23 | D | Quadrilaterals & Polygon Areas | Cut the octagon into 8 equal triangles from its center. The rectangle is made of 4 triangles, each equal in area to one of the 8. | |
| 24 | E | Sequences & Series | Each step adds -2y, so the 3rd and 4th terms are x - 3y and x - 5y. Dividing xy = x - 3y by y gives y = -3/5. | |
| 25 | B | Modular Arithmetic | The 90 numbers 1023, 1123, ..., 9923 go up by 100. Each step changes the remainder after dividing by 3, so every third number works. |
2003 AMC 10B. 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.