2018 AMC 12B. Problem 1
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1 2018 AMC 12B Problem 1 Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain? Problem 2 Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes? Problem 3 A line with slope 2 intersects a line with slope 6 at the point between the -intercepts of these two lines?. What is the distance Problem 4 A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle? chiefmathtutor@gmail.com Page 1
2 Problem 5 How many subsets of contain at least one prime number? Problem 6 Suppose cans of soda can be purchased from a vending machine for quarters. Which of the following expressions describes the number of cans of soda that can be purchased for dollars, where 1 dollar is worth 4 quarters? Problem 7 What is the value of Problem 8 Line Segment is a diameter of a circle with. Point, not equal to or, lies on the circle. As point moves around the circle, the centroid (center of mass) of (insert triangle symbol) traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve? chiefmathtutor@gmail.com Page 2
3 Problem 9 What is Problem 10 A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list? Problem 11 A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point in the figure on the right. The box has base length and height. What is the area of the sheet of wrapping paper? chiefmathtutor@gmail.com Page 3
4 Problem 12 Side of has length. The bisector of angle meets at, and. The set of all possible values of is an open interval. What is? Problem 13 Square has side length. Point lies inside the square so that and. The centroids of,,, and are the vertices of a convex quadrilateral. What is the area of that quadrilateral? chiefmathtutor@gmail.com Page 4
5 Problem 14 Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age? Problem 15 How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3? chiefmathtutor@gmail.com Page 5
6 Problem 16 The solutions to the equation are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled and. What is the least possible area of Problem 17 Let and be positive integers such that and is as small as possible. What is? Problem 18 A function is defined recursively by and for all integers. What is? Problem 19 Mary chose an even -digit number. She wrote down all the divisors of in increasing order from left to right:. At some moment Mary wrote as a divisor of. What is the smallest possible value of the next divisor written to the right of. chiefmathtutor@gmail.com Page 6
7 Problem 20 Let be a regular hexagon with side length. Denote,, and the midpoints of sides,, and, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of and? Problem 21 In with side lengths,, and, let and denote the circumcenter and incenter, respectively. A circle with center is tangent to the legs and and to the circumcircle of. What is the area of? Problem 22 Consider polynomials of degree at most, each of whose coefficients is an element of. How many such polynomials satisfy? Problem 23 Ajay is stading at point near Pontianak, Indonesia, latitude and longitude. Billy is standin at point near Big Baldy Mountain, Idaho, USA, latitude and longitude. chiefmathtutor@gmail.com Page 7
8 Assume that Earth is a perfect sphere with center. What is the degree measure of? Problem 24 How many satisfy the equation? Problem 25 Circles,, and each have radius and are placed in the plane so that each circle is externally tangent to the other two. Points,, and lie on,, and respectively such that and line is tangent to for each, where. See the figure below. The area of can be written in the form for positive integers and. What is? chiefmathtutor@gmail.com Page 8
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