2025 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 126 KB — a separate file from the paper, so one can be handed over without the other.

#	Answer	Topic	Key insight
1	E	Linear Equations & Word Problems	Only each rider's distance from home matters, and that is speed times riding time. Remember Betsy has ridden one hour less than Andy.
2	B	Ratios, Percents & Averages	Follow only the peanuts. The peanut pounds must be 40 percent of the new total weight; that tells how much mix was added, and then count the cashews.
3	D	Triangles: Area & Pythagorean	Split by which sides are equal. Two sides of 2025 works with any third side up to 2025. Two equal shorter sides must add to more than 2025.
4	A	Ratios, Percents & Averages	Turn each average into a total. Then Ash's age is 14 plus 2 per student, and also 52 minus 3 per teacher. Both agree at 7 students.
5	E	Sequences & Series	Break the sequence into groups that each end with 1, count how many terms the first few groups hold, then walk forward to position 2025.
6	C	Angles & Polygons	Each hexagon corner is where rays from two triangle corners meet. The small triangle they form with a side has base angles 20 and 20, or 40 and 40.
7	E	Polynomials	Dividing by x - 1 leaves the value at x = 1 as the remainder; same for x - 2. That gives two easy equations in a and b.
8	B	Logic Puzzles	Every statement only talks about how many statements are true. Try each possible count from 0 to 4 and keep the one that agrees with itself.
9	C	Functions	Sliding the graph sideways does not change its heights. So count how many times the original graph reaches height 25; each such spot gives one value of a.
10	C	Circles	The small radius equals the height of the chord above AB. A right triangle with legs 8 and that height gives big radius squared minus small radius squared = 64.
11	E	Sequences & Series	The geometric sequence 1, p, q, z makes z a perfect cube. The arithmetic one makes z one more than a multiple of 3. Test cubes 8, 27, 64.
12	D	Basic Counting	Sort digits 1 to 9 into groups: 2 is even and prime, 4, 6, 8 only even, 3, 5, 7 only prime, 1 and 9 neither. Then count two cases.
13	D	Sequences & Series	Each shaded ring is a shrunken copy of the one two rings further out, so the shaded areas add up to 1/(1 + k squared).
14	B	Basic Probability	The students sit side by side with chance 6/15. The four chairs left then form a row, not a circle, so the teachers have 3 good pairs out of 6.
15	A	Similar & Congruent Triangles	Draw AE. It is the hypotenuse of both right triangles ABE and ADE, so DE = 5. Then triangles ABC and EDC are similar, with EDC five times bigger.
16	D	Expected Value	List the 27 equally likely ways to drop the coins. Just 3 put all coins together and 6 spread them out; the other 18 give a fullest jar of 2.
17	E	GCD & LCM	Take away the remainders: N divides both 273420 and 272745, so it divides 45, their greatest common factor. A remainder of 16 forces N above 16, so N = 45.
18	B	Quadratics	For every factor, the reciprocals of its two roots add to -4/3, whatever k is, and the harmonic mean only needs these reciprocals.
19	A	Sequences & Series	Each row adds up to twice the row above, so 12,288 is the sum of row 13. Then follow only the second and third numbers down to that row.
20	A	Coordinate Geometry	Put the silo's center at (0, 0). The line of sight touches the silo when its distance from the center equals the radius, 10. That gives one equation for g.
21	C	Number Properties	Look at the largest number in the set. Pair up the smaller numbers so each pair adds to it; the set can hold at most one number from each pair.
22	B	Circles	The three outer centers are 3, 4 and 5 apart, a right triangle. Put it on coordinate axes and write the small circle's distance to each outer center.
23	D	Triangles: Area & Pythagorean	Use Pythagoras twice to find where the altitude from C meets AB. The bisector from B then cuts that altitude in the ratio of the sides beside angle B.
24	C	Basic Counting	A fair number's digits climb to the largest digit, then fall. Once the digits are chosen, every digit except the largest just picks the left or right side.
25	A	Geometric Probability	Whether AP beats BP depends on the square's vertical middle line; whether AP beats AB depends on a quarter circle around A. AP is the middle length in two regions.

2025 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.