2019 AIME II 真题

Problem 1

Two different points, C and D, lie on the same side of line AB so that ∆ABC and ∆BAD are congruent with AB = 9, BC = AD = 10, and CA = DB = 17. The intersection of these two triangular regions has area , where m and n are relatively prime positive integers. Find m + n.

Problem 2

Lily pads 1, 2, 3, ... lie in a row on a pond. A frog makes a sequence of jumps starting on pad . From any pad k the frog jumps to either pad k + 1 or pad k + 2 chosen randomly with probability  and independently of other jumps. The probability that the frog visits pad  is , where p and q are relatively prime positive integers. Find p + q.

Problem 3

Find the number of 7-tuples of positive integers (a, b, c, d, e, f, g) that satisfy the following system of equations: abc = 70cde = 71efg = 72

Problem 4

A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is , where m and n are relatively prime positive integers. Find m + n.

Problem 5

Four ambassadors and one advisor for each of them are to be seated at a round table with 12 chairs numbered in order 1 to 12. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are N ways for the 8 people to be seated at the table under these conditions. Find the remainder when N is divided by 1000.

Problem 6

In a Martian civilization, all logarithms whose bases are not specified as assumed to be base , for some fixed b ≥ 2. A Martian student writes down  and finds that this system of equations has a single real number solution x > 1. Find b.

Problem 7

Triangle ABC has side lengths AB = 120, BC = 220, and AC = 180. Lines , and  are drawn parallel to , and , respectively, such that the intersections of , and  with the interior of ∆ABC are segments of lengths 55, 45, and 15, respectively. Find the perimeter of the triangle whose sides lie on lines , and .

Problem 8

The polynomial f(z) = az2018 + bz2017 + cz2016 has real coefficients not exceeding 2019, and . Find the remainder when f(1) is divided by 1000.

Problem 9

Call a positive integer n k-pretty if n has exactly k positive divisors and n is divisible by k. For example, 18 is 6-pretty. Let S be the sum of the positive integers less than 2019 that are -pretty. Find .

Problem 10

There is a unique angle θ between 0° and 90° such that for nonnegative integers n, the value of tan(2nθ) is positive when n is a multiple of 3, and negative otherwise. The degree measure of θ is , where p and q are relatively prime positive integers. Find p + q.

Problem 11

Triangle ABC has side lengths AB = 7, BC = 8, and CA = 9. Circle 

ω1 passes through B and is tangent to line AC at A Circle ω2 passes through C and is tangent to line AB at A. Let K be the intersection of circles ω1 and ω2 not equal to A Then , where m and n are relatively prime positive integers. Find m + n.

Problem 12

For n ≥ 1 call a finite sequence (a1, a2, ..., an) of positive integers progressive if ai < ai+1  and ai divides ai+1 for 1 ≤ i ≤ n - 1. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to 360

Problem 13

Regular octagon A1A2A3A4A5A6A7A8  is inscribed in a circle of area 1. Point P lies inside the circle so that the region bounded by  and the minor arc  of the circle has area , while the region bounded by  and the minor arc  of the circle has area . There is a positive integer n such that the area of the region bounded by  and the minor arc  of the circle is equal to . Find n.

Problem 14

Find the sum of all positive integers n such that, given an unlimited supply of stamps of denominations 5, n, and n + 1 cents, 91 cents is the greatest postage that cannot be formed.

Problem 15

In acute triangle ABC points P and Q are the feet of the perpendiculars from C to  and from B to , respectively. Line PQ intersects the circumcircle of ∆ABC in two distinct points, X and Y. Suppose XP = 10, PQ = 25, and QY = 15. The value of AB ∙ AC can be written in the form  where m and n are positive integers, and n is not divisible by the square of any prime. Find m + n.