2024 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.

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

#	Answer	Topic	Key insight
1	B	Basic Counting	The person in the middle is counted once from each end. Count the people on each side of that person, then add that person back once.
2	B	Algebraic Manipulation	10 times 9 times 8 is 720, which is also 6!. So 10! and 7! times 6! are the same number, and their difference is zero.
3	E	Absolute Value & Inequalities	7 times pi is just under 22, so 2x can go from -21.99 to 21.99. That lets x be every whole number from -10 to 10.
4	D	Sequences & Series	Step 63 ends at ball 2016, so ball 2024 is in step 64. Bins repeat every 5 steps, and step 64 lands on bin D.
5	B	Sequences & Series	The full sum is 2500. Flipping a plus to a minus takes away twice that number, so the flipped numbers must add up to more than 1250.
6	B	Divisibility & Factors	The perimeter is smallest when the two sides are as close as possible. Since 2024 is one less than 45 squared, the sides 44 and 46 work.
7	A	Modular Arithmetic	Pull out the common factor 7 to the power 2024. What is left is 1 + 7 + 49 = 57, which is 3 times 19.
8	D	Divisibility & Factors	The divisors of 42 pair up so each pair multiplies to 42. Four pairs make N equal to 42 to the 4th power, which ends in 6.
9	A	Algebraic Manipulation	Squaring the sum of the numbers gives their squares plus twice their pairwise products. So 0 = 30 plus twice the products, and the products add to -15.
10	A	Similar & Congruent Triangles	Triangle AFE is a half-size copy of triangle CFB, so its area is one quarter as big. Give AFE area 1 and find every other piece from there.
11	C	Similar & Congruent Triangles	Equal areas make XM twice ZA. The right angle at M makes the two corner triangles WXM and MYA the same shape, which pins down ZA = 1.
12	A	Basic Counting	All students speak the same number of languages, so the rule just says no two students speak exactly the same group. Choosing 4 of 9 languages gives 126 groups.
13	B	Exponents, Logarithms & Radicals	The square root of 1183 is 13 root 7, so root x and root y are whole multiples of root 7 that add to 13.
14	B	Geometric Probability	The board is a tilted square of area 128. The target is a ring of area 14 pi that fits exactly inside it.
15	C	Statistics & Data	The mean makes x + y + z equal 11.2 or 20.2, and the median must be x, y or z. Then split by the smallest and largest numbers.
16	A	Games & Processes	Each move removes three numbers, so the count always leaves remainder 2 when divided by 3. The odd numbers never grow past 1012. Only 1010 fits both.
17	D	Arrangements with Restrictions	Split into cases by how many snails tie. Choose the tied snails, treat them as one block, then line up the block and the other snails.
18	B	Modular Arithmetic	Multiples of 5 give remainder 0. Any other number to the 100th power is 1 more than a multiple of 125, built up one factor of 5 at a time.
19	C	Coordinate Geometry	A line through two lattice points has a fraction slope. A fraction slope through one lattice point hits infinitely many. So each row has exactly two Possible entries.
20	A	Arrangements with Restrictions	A left shoe next to a right shoe must be its own mate. First find the allowed left-right patterns, then see which pairs are forced into place.
21	C	Circles	Two touching circles of radii r and s sit 2 times root(rs) apart along the floor. Measure along the floor for each spot the third pipe could take.
22	A	Basic Counting	Line up the 16 people, cut the line into four groups of 4, and divide out orders that do not matter. Then count the 3s on top and bottom.
23	B	Sequences & Series	The ten fractions come out to whole numbers 1, 3, 4, 7, 11, ... and each one is the sum of the two before it, just like Fibonacci numbers.
24	E	Modular Arithmetic	Put everything over 8. For even m each top term is a multiple of 8. For odd m the top becomes 4(m + 1), a multiple of 8.
25	E	Systems of Equations	A new side cannot grow from the same brick edge as before. That forces 7a = 3c + 1, 2b = 3a + 1, 2c = 3b + 1.

2024 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.