2026 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 133 KB — a separate file from the paper, so one can be handed over without the other.

#	Answer	Topic	Key insight
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. 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.