2020 AIME II 真题
2020 AIME II Problems
Problem 1
Find the number of ordered pairs of positive integers (m, n) such that m2n = 2020.
Problem 2
Let P be a point chosen uniformly at random in the interior of the unit square with vertices at (0, 0), (1, 0) (1, 1), and (0, 1). The probability that the slope of the line determined by P and the point 
is greater than or equal to 
can be written as 
, where m and n are relatively prime positive integers. Find m + n.
Problem 3
The value of x that satisfies 
can be written as 
, where m and n are relatively prime positive integers. Find m + n.
Problem 4
Triangles ∆ABC and ∆A′B′C′ lie in the coordinate plane with vertices A(0, 0), B(0, 12), C(16, 0), A′(24, 18), B′(36, 18), C′(24, 2). A rotation of m degrees clockwise around the point (x,y ) where 0 < m < 180, will transform ∆ABC to ∆A′B′C′. Find m + x + y .
Problem 5
For each positive integer n, let f(n) be the sum of the digits in the base-four representation of n and let g(n) be the sum of the digits in the base-eight representation of f(n). For example, f(2020) = f(1332104) = 10 = 128, and g(2020) = the digit sum of 128 = 3. Let N be the least value of n such that the base-sixteen representation of g(n) cannot be expressed using only the digits 0 through 9. Find the remainder when N is divided by 1000.
Problem 6
Define a sequence recursively by t1 = 20, t2 = 21, and 
for all n ≥ 3. Then t2020 can be written as 
, where p and q are relatively prime positive integers. Find p + q.
Problem 7
Two congruent right circular cones each with base radius 
and height 8 have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance 3 from the base of each cone. A sphere with radius r lies within both cones. The maximum possible value of r2 is 
, where m and n are relatively prime positive integers. Find m + n.
Problem 8
Define a sequence recursively by f1(x) = |x -1| and fn(x) = fn-1(|x - n|) for integers n > 1. Find the least value of n such that the sum of the zeros of fn exceeds 500,000.
Problem 9
While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.
Problem 10
Find the sum of all positive integers 
such that when 13 + 23 + 33 + ... + n3 is divided by n + 5, the remainder is 17.
Problem 11
Let P(x) = x2 - 3x - 7, and let Q(x) and R(x) be two quadratic polynomials also with the coefficient of x2 equal to 1. David computes each of the three sums P + Q, P + R, and Q + R and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If Q(0) = 2, then 
, where m and n are relatively prime positive integers. Find m + n.
Problem 12
Let m and n be odd integers greater than 1. An m x n rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers 
through n, those in the second row are numbered left to right with the integers n + 1 through 2n, and so on. Square 200 is in the top row, and square 2000 is in the bottom row. Find the number of ordered pairs (m, n) of odd integers greater than 
with the property that, in the m x n rectangle, the line through the centers of squares 200 and 2000 intersects the interior of square 1099.
Problem 13
Convex pentagon ABCDE has side lengths AB = 5, BC = CD = DE = 6, and EA = 7. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of ABCDE.
Problem 14
For real number x let 
be the greatest integer less than or equal to x, and define 
to be the fractional part of x. For example, {3} = 0 and {4.56} = 0.56. Define f(x) = x{x}, and let N be the number of real-valued solutions to the equation f(f(f(x))) = 17 for 0 ≤ x ≤ 2020. Find the remainder when N is divided by 1000.
Problem 15
Let ∆ABC be an acute scalene triangle with circumcircle ω. The tangents to ω at B and C intersect at T. Let X and Y be the projections of T onto lines AB and AC, respectively. Suppose BT = CT = 16, BC = 22, and TX2 + TY2 +XY2 = 1143. Find XY2.
