2020 AIME I 真题

2020 AIME I Problems

Problem 1

In ∆ABC with AB = AC, point D lies strictly between A and C on side ,  and point E lies strictly between A and B on side  such that AE = ED = DB = BC. The degree measure of ∠ABC is , where m and n are relatively prime positive integers. Find m+n.

Problem 2

There is a unique positive real number  such that the three numbers log8 2x, log4 x, and log2 x, in that order, form a geometric progression with positive common ratio. The number x can be written as , where m and n are relatively prime positive integers. Find m+n.

Problem 3

A positive integer N has base-eleven representation a b c and base-eight representation 1 b c a where a, b and c represent (not necessarily distinct) digits. Find the least such N expressed in base ten.

Problem 4 

Let S be the set of positive integers N with the property that the last four digits of N are 2020 and when the last four digits are removed, the result is a divisor of N For example, 42, 020 is in S because 4 is a divisor of 42, 020. Find the sum of all the digits of all the numbers in S For example, the number 420, 020 contributes 4 + 2 + 0 + 2 + 0 = 8 to this total.

Problem 5

Six cards numbered 1 through 6 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.

Problem 6

A flat board has a circular hole with radius 1 and a circular hole with radius 2 such that the distance between the centers of the two holes is 7 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 m + n

Problem 7

A club consisting of 11 men and 12 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 1 member or as many as 23 members. Let N be the number of such committees that can be formed. Find the sum of the prime numbers that divide N

Problem 8 

A bug walks all day and sleeps all night. On the first day, it starts at point O, faces east, and walks a distance of 5 units due east. Each night the bug rotates 60° counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point P Then  where  and  are relatively prime positive integers. Find m + n

Problem 9

Let S be the set of positive integer divisors of 209 Three numbers are chosen independently and at random with replacement from the set S and labeled a1, a2 and a3 in the order they are chosen. The probability that both a1 divides a2 and a2 divides a3 is , where m and n are relatively prime positive integers. Find m.

Problem 10

Let  and  be positive integers satisfying the conditions

l                      gcd (m + n, 210) = 1

l                      mm is a multiple of nn and

l                      m is not a multiple of n.

Find the least possible value of m + n.

Problem 11

For integers a, b, c and d, let f(x) = x2 + ax + b and g(x) = x2 +cx +d. Find the number of ordered triples (a, b, c) of integers with absolute values not exceeding 10 for which there is an integer d such that g(f(2)) = g(f(4)) = 0.

Problem 12

Let n be the least positive integer for which 149n - 2n is divisible by 33 ∙ 55 ∙ 77. Find the number of positive integer divisors of n.

Problem 13

Point D lies on side  of ∆ABC so that  bisects ∠BAC. The perpendicular bisector of  intersects the bisectors of ∠ABC and ∠ACB in points E and F, respectively. Given that AB = 4, BC = 5, and CA = 6, the area of ∆AEF can be written as , where m and p are relatively prime positive integers, and n is a positive integer not divisible by the square of any prime. Find m + n +p.

Problem 14

Let P(x) be a quadratic polynomial with complex coefficients whose x2 coefficient is 1. Suppose the equation P(P(x)) = 0 has four distinct solutions, x = 3, 4, a, b. Find the sum of all possible values of (a + b)2.

Problem 15

Let ∆ABC be an acute triangle with circumcircle ω, and let H be the intersection of the altitudes of ∆ABC. Suppose the tangent to the circumcircle of ∆HBC at H intersects ω at points X and Y with HA = 3, HX = 2, and HY = 6. The area of 

∆ABC can be written as , where m and n are positive integers, and n is not divisible by the square of any prime. Find m + n.