2019 AIME II Problems

2019 AIME II (Answer Key) | AoPS Contest Collections • PDF

Instructions This is a 15-question, 3-hour examination. All answers are integers ranging from to , inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers. No aids other than scratch paper, rulers and compasses are permitted. In particular, graph paper, protractors, calculators and computers are not permitted.

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Problem 1

Two different points, and , lie on the same side of line so that and are congruent with , and . The intersection of these two triangular regions has area , where and are relatively prime positive integers. Find .

Solution

Problem 2

Lily pads lie in a row on a pond. A frog makes a sequence of jumps starting on pad . From any pad the frog jumps to either pad or pad chosen randomly with probability and independently of other jumps. The probability that the frog visits pad is , where and are relatively prime positive integers. Find .

Solution

Problem 3

Find the number of -tuples of positive integers that satisfy the following system of equations:

Solution

Problem 4

A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is , where and are relatively prime positive integers. Find .

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

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Problem 8

Solution

Problem 9

Solution

Problem 10

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Problem 11

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Problem 12

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Problem 13

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Problem 14

Solution

Problem 15

Solution

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