Difference between revisions of "2018 AMC 12B Problems"
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Side <math>\overline{AB}</math> of <math>\triangle ABC</math> has length <math>10</math>. The bisector of angle <math>A</math> meets <math>\overline{BC}</math> at <math>D</math>, and <math>CD = 3</math>. The set of all possible values of <math>AC</math> is an open interval <math>(m,n)</math>. What is <math>m+n</math>? | Side <math>\overline{AB}</math> of <math>\triangle ABC</math> has length <math>10</math>. The bisector of angle <math>A</math> meets <math>\overline{BC}</math> at <math>D</math>, and <math>CD = 3</math>. The set of all possible values of <math>AC</math> is an open interval <math>(m,n)</math>. What is <math>m+n</math>? | ||
− | < | + | <math>\textbf{(A) }16 \qquad |
\textbf{(B) }17 \qquad | \textbf{(B) }17 \qquad | ||
\textbf{(C) }18 \qquad | \textbf{(C) }18 \qquad | ||
\textbf{(D) }19 \qquad | \textbf{(D) }19 \qquad | ||
− | \textbf{(E) }20 \qquad</ | + | \textbf{(E) }20 \qquad</math> |
[[2018 AMC 12B Problems/Problem 12|Solution]] | [[2018 AMC 12B Problems/Problem 12|Solution]] |
Revision as of 19:22, 28 October 2018
2018 AMC 12B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
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 . What is the distance between the -intercepts of these two lines?
Problem 4
A circle has a chord of length , and the distance from the center of the circle to the chord is . What is the area of the circle?
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 traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
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?
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?
Problem 14
Joey, 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?
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 ?
Problem 20
Let be a regular hexagon with side length . Denote by , , 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 standing at point near Pontianak, Indonesia, latitude and longitude. Billy is standing at point near Big Baldy Mountain, Idaho, USA, latitude and longitude. Assume that Earth is a perfect sphere with center . What is the degree measure of ?
Problem 24
Let denote the greatest integer less than or equal to . How many real numbers 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 ?
See also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2018 AMC 12A Problems |
Followed by 2019 AMC 12A Problems |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.