2021 AIME I 真题
2021 AIME I Problems
Problem 1
Zou and Chou are practicing their 100-meter sprints by running 
races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is 
if they won the previous race but only 
if they lost the previous race. The probability that Zou will win exactly 5 of the 6 races is 
, where m and n are relatively prime positive integers. Find m + n.
Problem 2
In the diagram below, ABCD is a rectangle with side lengths AB = 3 and BC = 11, and AECF is a rectangle with side lengths AF = 7 and FC = 9, as shown. The area of the shaded region common to the interiors of both rectangles is 
, where m and n are relatively prime positive integers. Find m + n.
Problem 3
Find the number of positive integers less than 1000 that can be expressed as the difference of two integral powers of 2
Problem 4
Find the number of ways 66 identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.
Problem 5
Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.
Problem 6
Segments 
and 
are edges of a cube and 
is a diagonal through the center of the cube. Point P satisfies 
and 
. What is PA?
Problem 7
Find the number of pairs (m, n) of positive integers with 1 ≤ m < n ≤ 30 such that there exists a real number x satisfying sin(mx) + sin(nx) = 2.
Problem 8
Find the number of integers 
such that the equation 
has 12 distinct real solutions.
Problem 9
Let ABCD be an isosceles trapezoid with AD = BC and AB < CD. Suppose that the distances from A to the lines BC, CD, and BD are 15, 18, and 10 respectively. Let K be the area of ABCD. Find 
Problem 10
Consider the sequence (ak)k≥1 of positive rational numbers defined by 
and for k ≥ 1, if 
for relatively prime positive integers m and n, then 
.
Determine the sum of all positive integers j such that the rational number aj can be written in the form 
for some positive integer t.
Problem 11
Let ABCD be a cyclic quadrilateral with AB = 4, BC = 5, CD = 6, and DA = 7. Let A1 and C1 be the feet of the perpendiculars from A and C, respectively, to line BD and let B1 and D1 be the feet of the perpendiculars from B and D respectively, to line AC. The perimeter of A1B1C1D1 is 
, where m and n are relatively prime positive integers. Find m + n.
Problem 12
Let A1A2A3...A12 be a dodecagon (12-gon). Three frogs initially sit at A4, A8, and A12. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is 
, where m and n are relatively prime positive integers. Find m + n.
Problem 13
Circles ω1 and ω2 with radii 961 and 625, respectively, intersect at distinct points A and B. A third circle ω is externally tangent to both ω1 and ω2. Suppose line AB intersects ω at two points P and Q such that the measure of minor arc 
is 120°. Find the distance between the centers of ω1 and ω2.
Problem 14
For any positive integer a, σ(a) denotes the sum of the positive integer divisors of a. Let n be the least positive integer such that σ(an) - 1 is divisible by 2021 for all positive integers a. Find the sum of the prime factors in the prime factorization of n.
Problem 15
Let S be the set of positive integers k such that the two parabolas y = x2 - k and x = 2(y - 20)2 - k intersect in four distinct points, and these four points lie on a circle with radius at most 21. Find the sum of the least element of S and the greatest element of S.
