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2026 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2026 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | B | Ratios, Percents & Averages — Measure every record in EP-lengths. One LP counts as 3 EPs, so the 33 LPs play as long as 99 EPs. |
| 2 | A | Ratios, Percents & Averages — The 27 percent is figured on the price before tax. The customer pays 127 percent of that price, so the tax is 27 parts out of 127. |
| 3 | D | Triangles: Area & Pythagorean — Base BE is 12 and lies on the bottom side, so only the height of F matters. The two lines cross 8 units above the bottom. |
| 4 | A | Angles & Polygons — A 150 and a 190 average out to 170 degrees, so the angle sum is 170 per side. Setting that equal to 180 times (sides minus 2) gives 36 sides. |
| 5 | C | Sequences & Series — Adding a new number equal to the current average leaves the average unchanged. So once the third term is 142, every later term is 142 too. |
| 6 | A | Circles — 16-30-34 is a right triangle, so its longest side is a diameter. That diameter splits the circle into two halves, and the triangle sits inside one half. |
| 7 | A | Systems of Equations — Subtracting the equations removes x and leaves y + z = 5. Then x must be 26, and only the pairs of positive whole numbers adding to 5 remain. |
| 8 | C | Linear Equations & Word Problems — To make the needed speed as small as possible, take the shortest distance that rounds to 7 miles and the longest time that rounds to 5 minutes. |
| 9 | D | Similar & Congruent Triangles — Enlarging makes every side longer, so the big triangle's 9 must come from the small triangle's third side, and its 12 from the small triangle's 9. |
| 10 | C | Statistics & Data — Use the sum-of-squares formula for the mean and the middle square for the median, both in terms of n, then make the mean 1.32 times the median. |
| 11 | D | GCD & LCM — The least common multiple of 2, 5, 24, 60 and 240 is 240. So either n is a multiple of 240, or n is a divisor of 240. |
| 12 | D | Basic Probability — Picture the socks in a random line. The socks left over are the unbroken same-color run at the end, so read the line from the back. |
| 13 | C | Basic Counting — Within-club plus between-club handshakes is all handshakes: 231, so 22 people. Between-club handshakes are the product of the two club sizes. |
| 14 | B | Triangles: Area & Pythagorean — Write all three triangle inequalities. Each becomes a quadratic inequality in t that factors, and together they trap t strictly between 2 and 10. |
| 15 | B | Games & Processes — Work upward. The node 20/27 has a partner 7/27 (they add to 1), and their parent is 7/20. Repeat until you reach 1/2, counting the steps. |
| 16 | D | Basic Probability — Balls beyond a color's cap score nothing. Waste as many as possible: the lowest score, 4, needs either no reds or all six reds. |
| 17 | C | Quadrilaterals & Polygon Areas — Each square reaches 1 unit inward, but the center is only √3/2 inward, so the small hexagon's center-to-side distance is 1 − √3/2. |
| 18 | D | Sequences & Series — Measure every term by how far it sits above 4. That distance is cut in half at each step, and it must stay a whole number through ten halvings. |
| 19 | C | Solid Geometry — The cover's center sits on the table and the top corners are hardest to reach. Find the smallest circle around the footprint, then add the height 2 by Pythagoras. |
| 20 | C | Distributions & Stars and Bars — 510,510 is the product of the seven primes 2 through 17, each used once. A box is just a way to split these seven primes into three non-empty groups. |
| 21 | C | Paths & Grids — Two scoring sections need all four X's, two in each, so the third section is all O's. Only two all-O sections exist; count the partitions around each. |
| 22 | A | Coordinate Geometry — Half a turn about the point halfway between two centers swaps those two equal ellipses, so if they touch, they touch exactly at that halfway point. |
| 23 | D | Number Properties — Every 6 consecutive integers hold exactly 3 even numbers and 2 multiples of 3, so only the leftover 0 to 5 numbers decide which pairs appear. |
| 24 | D | Polynomials — Subtract the two values. The constant cancels, and what remains depends only on the inputs' sum 6 and product 4, forcing c = −32 − 6b. |
| 25 | B | Games & Processes — Room 1 gains at most one nut a day while the average gains 0.9, so it catches up only 0.1 a day; starting 9.5 behind takes 95 days. |
2026 AMC 10A — answer key. 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.