2024 AMC 12B Problems

2024 AMC 12B (Answer Key) Printable versions: • AoPS Resources • PDF

Instructions This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.

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Problem 1

In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in line?

Solution

Problem 2

What is ?

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Problem 3

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Problem 4

Balls numbered are deposited in bins, labeled and , using the following procedure. Ball is deposited in bin , and balls and are deposited in . The next three balls are deposited in bin , the next in bin , and so on, cycling back to bin after balls are deposited in bin . (For example, are deposited in bin at step 7 of this process.) In which bin is ball deposited?

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Problem 5

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Problem 6

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Problem 7

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Problem 8

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Problem 9

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Problem 10

A list of 9 real numbers consists of , , , , , , as well as with . The range of the list is , and the mean and median are both positive integers. How many ordered triples are possible?

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Problem 11

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Problem 12

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Problem 13

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Problem 14

How many different remainders can result when the th power of an integer is divided by ?

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Problem 15

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Problem 16

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Problem 17

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Problem 18

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Problem 19

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Problem 20

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Problem 21

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Problem 22

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Problem 23

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Problem 24

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Problem 25

Pablo will decorate each of identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the decisions he must make. After the paint dries, he will place the balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as where and are relatively prime positive integers. What is (Recall that two events and are independent if )

Solution

See also

AMC 12

AMC 12 Problems and Solutions

Mathematics competitions

Mathematics competition resources