2009 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 126 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | B | Linear Equations & Word Problems | Five muffins cost 2 dollars and 50 cents, and each swap to a bagel adds 25 cents, so two swaps reach a whole dollar amount. | |
| 2 | C | Fractions & Decimals | Work out the top and the bottom separately first: the top is 1/12 and the bottom is 1/6, and 1/12 is half of 1/6. | |
| 3 | C | Ratios, Percents & Averages | The three lost cans were exactly the paint for the 5 rooms she can no longer do, so 3 cans paint 5 rooms, and 25 rooms need 15 cans. | |
| 4 | C | Quadrilaterals & Polygon Areas | The two triangles use up 25 - 15 = 10 meters of the top edge, so each has legs of 5, which is also the yard's width. | |
| 5 | D | Ratios, Percents & Averages | Twenty percent less than 60 is 48. A number plus a third of itself is 4 thirds of it, so 4 thirds make 48. | |
| 6 | D | Divisibility & Factors | The twins share an age, so 128 is Kiana's age times the twins' age twice. Try the few possible twin ages and keep the one where Kiana is younger. | |
| 7 | C | Basic Counting | Only the single plus sign can be moved earlier. It can join the 3 with the left product, the right product, both, or neither, giving 26, 46, 50 and 70. | |
| 8 | B | Ratios, Percents & Averages | Follow a price of 100 dollars month by month. It reaches 120 after March, and dropping back to 100 is 20 out of 120, about 17 percent. | |
| 9 | A | Angles & Polygons | The equal sides meet at B, so triangle ABC has angles B, and twice five-halves of B, giving B = 30. Vertical angles carry C's angle across. | |
| 10 | E | Triangles: Area & Pythagorean | The stump, the ground, and the broken top form a right triangle with legs x and 1 and long side 5 - x. When you expand, the x-squared terms cancel. | |
| 11 | A | Basic Counting | The middle digit has no partner, so it must be the 5 that appears an odd number of times. The first three digits are 2, 3, 5 in any order. | |
| 12 | A | Triangles: Area & Pythagorean | Every triangle has two corners on one line and one on the other, so its height never changes. Only the base length matters: 1, 2 or 3. | |
| 13 | C | Modular Arithmetic | Rolling lays the sides down in order AB, BC, CD, DE, EA, repeating every 23 units. 2009 leaves remainder 8 when divided by 23, landing on CD. | |
| 14 | D | Sequences & Series | After each refill there is always exactly 3/4 quart of other seeds, since the birds eat them all. Track only the millet until it passes 3/4 quart. | |
| 15 | E | Linear Equations & Word Problems | Going from half full to two-thirds full adds one-sixth of the water, and that one-sixth weighs a - b. Add the last one-third (twice that) to the two-thirds weighing. | |
| 16 | B | Circles | BO cuts the 60 degree angle at B in half, and radius OA meets the tangent at a right angle, so triangle OAB is 30-60-90 with BO twice the radius. | |
| 17 | C | Coordinate Geometry | The shaded part is the big right triangle with corners (c,0), (3,0), (3,3), minus the one empty unit square in its corner. Set it equal to 5/2. | |
| 18 | D | Similar & Congruent Triangles | Right triangles AME and ABC share the angle at A, so they have the same shape, and ME is 3/4 of AM just as BC is 3/4 of AB. | |
| 19 | A | Basic Counting | The time is right only when no 1 appears. Multiply the share of good hours (8 of 12) by the share of good minutes (45 of 60). | |
| 20 | B | Triangle Centers & Cevians | An angle bisector splits the opposite side in the same ratio as the two sides that form the angle: here 1 to the square root of 5. | |
| 21 | D | Modular Arithmetic | Divided by 8, powers of 3 leave remainders 1, 3, 1, 3, and so on. Each pair adds to 4, and there are 1005 pairs. | |
| 22 | B | Solid Geometry | The cuts are vertical, so the piece is triangle B stretched to height 2. Cake is twice B's area; icing is B's area plus a 2 by 2 side. | |
| 23 | C | Geometric Probability | A runner is in the photo within one-eighth of a lap time of crossing the start line. Find each runner's time window and see where they overlap. | |
| 24 | A | Angles & Polygons | Extend the slanted sides of all the trapezoids down to the center of the arch. The nine trapezoids split the 180 degree straight angle into nine 20 degree slices. | |
| 25 | B | Basic Probability | A loop must run through four faces forming a belt. There are 3 belts; each needs its 4 faces right, the other 2 faces are free, and belts never overlap. |
2009 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.