2023 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 125 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	Each hour the two riders get 18 + 12 = 30 miles closer, so they meet after 1.5 hours. Then find how far Alicia rode.
2	A	Linear Equations & Word Problems	Take away what the two sides share. What is left says 5/12 of a pizza weighs the same as 3 cups of orange slices, which is 3/4 pound.
3	A	Number Properties	A square is a multiple of 5 only when the number squared is a multiple of 5. Count 5, 10, 15, and so on while the square stays below 2023.
4	D	Geometric Optimization	The longest side must be shorter than the other three sides added together. With a perimeter of 26 that means it is less than 13, and 12 works.
5	E	Exponents, Logarithms & Radicals	Break every number into primes. Pair each 2 with a 5 to make 10s. What is left is 243 followed by fifteen zeros.
6	D	Basic Counting	Follow one vertex's number: 3 edges use it and each edge sits on 2 faces, so it is counted 6 times. The cube's value is 6 times 21.
7	B	Basic Probability	Each roll adds at least 1, so the total can reach 3 only in the first three rolls. List the orders adding to 3: 3; 1,2; 2,1; 1,1,1.
8	D	Linear Equations & Word Problems	200 is 90 of the 240 degrees from 110 to 350, which is 3/8 of the way. On the new scale that is 3/8 of 100, or 37.5.
9	E	Bases & Digits	The year 2023 already has one 0 and one 3. So the month and day together must use one 0, one 3, and two copies of some other digit.
10	D	Ratios, Percents & Averages	Turn each mean into a total: n quizzes with mean m add up to n times m. Each story then gives a simple equation.
11	C	Triangles: Area & Pythagorean	The legs add to the big square's side and their squares add to 2. That gives their product, and sum plus product gives the ratio.
12	B	Divisibility & Factors	Reversing N moves its first digit to the end. A multiple of 5 ends in 0 or 5, and N cannot start with 0, so N is in the 500s.
13	C	Circles	All the spots where Bharat sees Abdul and Chiang at a 60 degree angle lie on one circle through them. Abdul-to-Bharat is longest when it is a diameter.
14	B	Conditional Probability & States	Only the 9 multiples of 11 up to 99 can work. For each, exactly half its divisors are multiples of 11, because each divisor d pairs with 11 times d.
15	E	Sequences & Series	Each shaded ring is the difference of two circles, and those areas are 3, 7, 11, ... times pi. With n rings the total is n(2n + 1) times pi.
16	B	Games & Processes	Every game has exactly one winner, so total wins equal total games. Left wins are 1.4 times right wins, which makes the total a multiple of 12.
17	A	Triangles: Area & Pythagorean	Go around the rectangle one right triangle at a time. Leg 30 forces 16-30-34; then leg 12 allows 5, 9 or 16, and only 9 makes the last triangle work.
18	D	Solid Geometry	Count edges from the faces (12 times 4, halved), then vertices from Euler's formula. Counting edge ends two ways gives the split into 3-edge and 4-edge vertices.
19	E	Coordinate Geometry	A rotation keeps each point the same distance from the center. So the center is on the perpendicular bisector of A and A', and of B and B'.
20	D	Paths & Grids	The center square is in all four 2 by 2 blocks, so pick it first. The four edge-middle squares then use the other three colors, and each corner is forced.
21	D	Polynomials	Each clue is a value of P: P(1) is 1, and P is 0 at 0, 4 and 9. Those roots alone make P(1) equal 24, so add another root.
22	D	Circles	C3 sits at the middle point between the two centers with radius 3/4. By symmetry C4's center is straight above that point, which gives one right triangle.
23	C	Diophantine Equations	N equals a(a + 20) and b(b + 23). If a were b + 2 or more, the first product would be too big, so a is b + 1.
24	C	Quadrilaterals & Polygon Areas	Only the frame's side matters. Neighboring blocks share a side line, which forces the two gaps on each frame edge to add up to 2, so the side is 3.
25	A	Basic Probability	Stand at R and sort the other 11 vertices by distance: 5 at distance 1, 5 at distance 2, 1 at distance 3. Then count pairs.

2023 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.