Difference between revisions of "2013 AMC 12B Problems"
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==Problem 25== | ==Problem 25== | ||
− | + | Let <math>G</math> be the set of polynomials of the form | |
+ | <cmath> P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50, </cmath> | ||
+ | where <math> c_1,c_2,\cdots, c_{n-1} </math> are integers and <math>P(z)</math> has distinct roots of the form <math>a+ib</math> with <math>a</math> and <math>b</math> integers. How many polynomials are in <math>G</math>? | ||
− | < | + | <math> \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D}}\ 992\qquad\textbf{(E)}\ 1056 </math> |
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− | |||
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[[2013 AMC 12B Problems/Problem 25|Solution]] | [[2013 AMC 12B Problems/Problem 25|Solution]] |
Revision as of 14:47, 22 February 2013
Current Progress: finished 15 problems. Onwards are all bullsh*t from previous years' problems.
2013 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
On a particular January day, the high temperature in Lincoln, Nebraska, was degrees higher than the low temperature, and the average of the high and low temperatures was . In degrees, what was the low temperature in Lincoln that day?
Problem 2
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is steps by steps. Each of Mr. Green's steps is feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?
Problem 3
When counting from to , is the number counted. When counting backwards from to , is the number counted. What is ?
Problem 4
Ray's car averages miles per gallon of gasoline, and Tom's car averages miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?
Problem 5
The average age of fifth-graders is . The average age of of their parents is . What is the average age of all of these parents and fifth-graders?
Problem 6
Real numbers and satisfy the equation . What is ?
Problem 7
Jo and Blair take turns counting from 1 to one more than the last number said by the other person. Jo starts by saing "1", so Blair follows by saying "1, 2". Jo then says "1, 2, 3", and so on. What is the number said?
Problem 8
Line has equation and goes through . Line has equation and meets line at point . Line has positive slope, goes through point , and meets at point . The area of is 3. What is the slope of ?
Problem 9
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides ?
Problem 10
Alex has red tokens and blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
Problem 11
Two bees start at the same spot and fly at the same rate in the following directions. Bee travels foot north, then foot east, then foot upwards, and then continues to repeat this pattern. Bee travels foot south, then foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly feet away from each other?
east, west
north, south
north, west
up, south
up, west
Problem 12
Cities , , , , and are connected by roads , , , , , , and . How many different routes are there from to that use each road exactly once? (Such a route will necessarily visit some cities more than once.)
Problem 13
The internal angles of quadrilateral form an arithmetic progression. Triangles and are similar with and . Moreover, the angles in each of these two triangles also form an arithemetic progression. In degrees, what is the largest possible sum of the two largest angles of ?
Problem 14
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is . What is the smallest possible value of ?
Problem 15
the number is expressed in the form
where and are positive integers and is as small as possible. What is ?
Problem 16
Rhombus has side length and . Region consists of all points inside of the rhombus that are closer to vertex than any of the other three vertices. What is the area of ?
Problem 17
Let , and for integers . What is the sum of the digits of ?
Problem 18
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
Problem 19
A lattice point in an -coordinate system is any point where both and are integers. The graph of passes through no lattice point with for all such that . What is the maximum possible value of ?
Problem 20
Triangle has , and . The points , and are the midpoints of , and respectively. Let be the intersection of the circumcircles of and . What is ?
Problem 21
The arithmetic mean of two distinct positive integers and is a two-digit integer. The geometric mean of and is obtained by reversing the digits of the arithmetic mean. What is ?
Problem 22
Let be a triangle with sides , and . For , if and , and are the points of tangency of the incircle of to the sides , and , respectively, then is a triangle with side lengths , and , if it exists. What is the perimeter of the last triangle in the sequence ?
Problem 23
A bug travels in the coordinate plane, moving only along the lines that are parallel to the -axis or -axis. Let and . Consider all possible paths of the bug from to of length at most . How many points with integer coordinates lie on at least one of these paths?
Problem 24
Let . What is the minimum perimeter among all the -sided polygons in the complex plane whose vertices are precisely the zeros of ?
Problem 25
Let be the set of polynomials of the form where are integers and has distinct roots of the form with and integers. How many polynomials are in ?
$\textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D}}\ 992\qquad\textbf{(E)}\ 1056$ (Error compiling LaTeX. Unknown error_msg)
See also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2013 AMC 12A Problems |
Followed by 2014 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 |