2017 AIME II Problems
2017 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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Contents
Problem 1
Find the number of subsets of that are subsets of neither nor .
Problem 2
Theams , , , and are in the playoffs. In the semifinal matches, plays , and plays . The winners of those two matches will play each other in the final match to determine the champion. When plays , the probability that wins is , and the outcomes of all the matches are independent. The probability that will be the champion is , where and are relatively prime positive integers. Find .
Problem 3
A triangle has vertices , , and . The probability that a randomly chosen point inside the triangle is closer to vertex than to either vertex or vertex can be written as , where and are relatively prime positive integers. Find .
Problem 4
Find the number of positive integers less than or equal to whose base-three representation contains no digit equal to .
Problem 5
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are , , , , , and . Find the greatest possible value of .
Problem 6
Find the sum of all positive integers such that is an integer.
Problem 7
Find the number of integer values of in the closed interval for which the equation has exactly one real solution.
Problem 8
Find the number of positive integers less than such that is an integer.
Problem 9
A special deck of cards contains cards, each labeled with a number from to and colored with one of seven solors. 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 cardf with each number, the probability that Sharon can discard one of her cards and have at least one card of each color and at least one card with each number if , where and are relatively prime positive integers. Find .
Problem 10
Rectangle has side lengths and . Point is the midpoint of , point is the trisection point of closer to , and point is the intersection of and . Point lies on the quadrilateral , and bisects the area of . Find the area of .
Problem 11
Five towns are connected by a system of raods. 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 has radius , and the point is a point on the circle. Circle has radius and is internally tangent to at point . Point lies on circle so that is located counterclockwise from on . Circle has radius and is internally tangent to at point . In this way a sequence of circles and a sequence of points on the circles are constructed, where circle has radius and is internally tangent to circle at point , and point lies on counterclockwise from point , as shown in the figure below. There is one point inside all of these circles. When , the distance from the center to is , where and are relatively prime positive integers. Find .
\usepackage{asymptote} \begin{figure}[h] \begin{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("",(125,0),E); dot((25,100)); label("",(25,100),SE); dot((-55,20)); label("",(-55,20),E); \end{asy} \end{figure}
Problem 13
For each integer , let be the number of -element subsets of the vertices of the regular -gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of such that .
Problem 14
A grid of points consists of all points in space of the form , where , , and are integers between and , inclusive. Find the number of different lines that contain exactly of these points.
Problem 15
Tetrahedron has , , and . For any point in space, define . The least possible value of can be expressed as , where and are positive integers, and is not divisible by the square of any prime. Find .