Problem Set Workbook
Access the downloadable workbook for 2020 AIME I problems here.
Discussion Forum
Engage in discussion about the 2020 AIME I math contest by visiting Random Math AIME I 2020 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2020 AIME I problems, please refer below:
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 .
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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.
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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: Six cards numbered through are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order.
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Problem 6: A flat board has a circular hole with radius and a circular hole with radius such that the distance between the centers of the two holes is . Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is , where and are relatively prime positive integers. Find .
Solution:
Problem 7: A club consisting of men and women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as member or as many as members. Let be the number of such committees that can be formed. Find the sum of the prime numbers that divide .
Solution:
Problem 8: A bug walks all day and sleeps all night. On the first day, it starts at point , faces east, and walks a distance of units due east. Each night the bug rotates counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to point . Then , where and are relatively prime positive integers. Find .
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Problem 9: Let be the set of positive integer divisors of . Three numbers are chosen independently and at random with replacement from the set and labeled , and in the order they are chosen. The probability that both divides and divides is , where and are relatively prime positive integers. Find .
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Problem 10: Let and be positive integers satisfying the conditions
,
is a multiple of , and
is not a multiple of .
Find the least possible value of .
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Problem 11: For integers , and , let and . Find the number of ordered triples of integers with absolute values not exceeding for which there is an integer such that .
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Problem 12: Let be the least positive integer for which is divisible by . Find the number of positive integer divisors of .
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Problem 13: Point lies on side of so that bisects . The perpendicular bisector of intersects the bisectors of and in points and , respectively. Given that , and , the area of can be written as , where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. Find .
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Problem 14: Let be a quadratic polynomial with complex coefficients whose coefficient is . Suppose the equation has four distinct solutions, . Find the sum of all possible values of .
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Problem 15: Let be an acute triangle with circumcircle , and let be the intersection of the altitudes of . Suppose the tangent to the circumcircle of at intersects at points and with , and . The area of can be written as , where and are positive integers, and is not divisible by the square of any prime. Find .
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions