2020 AIME I Problems

2020 AIME I (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, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.

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Problem 1

In with point lies strictly between and on side and point lies strictly between and on side such that The degree measure of is where and are relatively prime positive integers. Find

Solution

Problem 2

There is a unique positive real number such that the three numbers and in that order, form a geometric progression with positive common ratio. The number can be written as where and are relatively prime positive integers. Find

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Problem 3

A positive integer has base-eleven representation and base-eight representation where and represent (not necessarily distinct) digits. Find the least such expressed in base ten.

Solution

Problem 4

Let be the set of positive integers with the property that the last four digits of are and when the last four digits are removed, the result is a divisor of For example, is in because is a divisor of Find the sum of all the digits of all the numbers in For example, the number contributes to this total.

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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

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Problem 11

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Problem 12

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Problem 13

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Problem 14

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Problem 15

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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.