2017 AIME II 真题

2017 AIME II Problems

Problem 1

Find the number of subsets of {1, 2, 3, 4, 5, 6, 7, 8} that are subsets of neither {1, 2, 3, 4, 5} nor {4, 5, 6, 7, 8} .

Problem 2

Teams T1, T2, T3, and T4 are in the playoffs. In the semifinal matches, T1 plays T4, and T2 plays T3. The winners of those two matches will play each other in the final match to determine the champion. When Ti plays Tj, the probability that Ti wins is , and the outcomes of all the matches are independent. The probability that T4 will be the champion is , where p and q are relatively prime positive integers. Find p + q.

Problem 3

A triangle has vertices A(0, 0), B(12, 0), and C(8, 10). The probability that a randomly chosen point inside the triangle is closer to vertex B than to either vertex A or vertex C can be written as , where p and q are relatively prime positive integers. Find p + q.

Problem 4

Find the number of positive integers less than or equal to 2017 whose base-three representation contains no digit equal to 0.

Problem 5

A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are 189, 320, 287, 234, x, and y. Find the greatest possible value of x + y.

Problem 6

Find the sum of all positive integers n such that $\sqrt{n^2+85n+2017}$ is an integer.

Problem 7

Find the number of integer values of k in the closed interval for which the equation $\log(kx)=2\log(x+2)$ has exactly one real solution.

Problem 8

Find the number of positive integers n less than 2017 such that $1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!}$ is an integer.

Problem 9

A special deck of cards contains 49 cards, each labeled with a number from 1 to 7 and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and still have at least one card of each color and at least one card with each number is , where p and q are relatively prime positive integers. Find p + q.

Problem 10

Rectangle ABCD has side lengths AB = 84 and AD = 42. Point M is the midpoint of $\overline{AD}$ , point is the trisection point of $\overline{AB}$ closer to A, and point is the intersection of $\overline{CM}$ and $\overline{DN}$ . Point P lies on the quadrilateral BCON, and $\overline{BP}$ bisects the area of BCON. Find the area of ∆CDP.

Problem 11

Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).

Problem 12

Circle C0 has radius 1, and the point A0 is a point on the circle. Circle C1 has radius r < 1 and is internally tangent to C0 at point A0. Point A1 lies on circle C1 so that A1 is located 90° counterclockwise from A0 on C1. Circle C2 has radius r2 and is internally tangent to C1 at point A1. In this way a sequence of circles C1, C2, C3, ... and a sequence of points on the circles A1, A2, A3, ... are constructed, where circle Cn has radius and is internally tangent to circle Cn-1 at point An-1, and point An lies on Cn90° counterclockwise from point An-1, as shown in the figure below. There is one point B inside all of these circles. When , the distance from the center C0 to B is , where m and n are relatively prime positive integers. Find m + n.

[asy] draw(Circle((0,0),125)); draw(Circle((25,0),100)); draw(Circle((25,20),80)); draw(Circle((9,20),64)); dot((125,0)); label("$A_0$",(125,0),E); dot((25,100)); label("$A_1$",(25,100),SE); dot((-55,20)); label("$A_2$",(-55,20),E); [/asy]

Problem 13

For each integer n ≥ 3, let f(n) be the number of 3-element subsets of the vertices of a regular n-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of such that f(n + 1) = f(n) + 78.

Problem 14

A 10 x 10 x 10 grid of points consists of all points in space of the form (i, j, k), where i, j, and k are integers between 1 and 10, inclusive. Find the number of different lines that contain exactly 8 of these points.

Problem 15

Tetrahedron ABCD has AD = BC = 28, AC = BD = 44, and AB = CD = 52. For any point X in space, define f(X) = AX + BX + CX + DX. The least possible value of f(X) can be expressed as $m\sqrt{n}$ , where m and n are positive integers, and n is not divisible by the square of any prime. Find m + n.