2017 AIME I Problems

2017 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

Fifteen distinct points are designated on : the 3 vertices , , and ; other points on side ; other points on side ; and other points on side . Find the number of triangles with positive area whose vertices are among these points.

Solution

Problem 2

When each of 702, 787, and 855 is divided by the positive integer , the remainder is always the positive integer . When each of 412, 722, and 815 is divided by the positive integer , the remainder is always the positive integer . Fine .

Solution

Problem 3

For a positive integer , let be the units digit of . Find the remainder when is divided by .

Solution

Problem 4

A pyramid has a triangular base with side lengths , , and . The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length . The volume of the pyramid is , where and are positive integers, and is not divisible by the square of any prime. Find .

Solution

Problem 5

A rational number written in base eight is , where all digits are nonzero. The same number in base twelve is . Find the base-ten number .

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

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