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All the roots of the polynomial <math>z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16</math> are positive integers, possibly repeated. What is the value of <math>B</math>? | All the roots of the polynomial <math>z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16</math> are positive integers, possibly repeated. What is the value of <math>B</math>? | ||
− | <math>\textbf{(A)} | + | <math>\textbf{(A) }{-}88 \qquad \textbf{(B) }{-}80 \qquad \textbf{(C) }{-}64 \qquad \textbf{(D) }{-}41\qquad \textbf{(E) }{-}40</math> |
[[2021 AMC 10A Problems/Problem 14|Solution]] | [[2021 AMC 10A Problems/Problem 14|Solution]] |
Latest revision as of 17:28, 4 December 2021
2021 AMC 10A (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
What is the value of
Problem 2
Portia's high school has times as many students as Lara's high school. The two high schools have a total of students. How many students does Portia's high school have?
Problem 3
The sum of two natural numbers is . One of the two numbers is divisible by . If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
Problem 4
A cart rolls down a hill, travelling inches the first second and accelerating so that during each successive -second time interval, it travels inches more than during the previous -second interval. The cart takes seconds to reach the bottom of the hill. How far, in inches, does it travel?
Problem 5
The quiz scores of a class with students have a mean of . The mean of a collection of of these quiz scores is . What is the mean of the remaining quiz scores in terms of ?
Problem 6
Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?
Problem 7
Tom has a collection of snakes, of which are purple and of which are happy. He observes that
- all of his happy snakes can add,
- none of his purple snakes can subtract, and
- all of his snakes that can't subtract also can't add.
Which of these conclusions can be drawn about Tom's snakes?
Purple snakes can add.
Purple snakes are happy.
Snakes that can add are purple.
Happy snakes are not purple.
Happy snakes can't subtract.
Problem 8
When a student multiplied the number by the repeating decimal, where and are digits, he did not notice the notation and just multiplied times Later he found that his answer is less than the correct answer. What is the -digit number
Problem 9
What is the least possible value of for real numbers and ?
Problem 10
Which of the following is equivalent to
Problem 11
For which of the following integers is the base- number not divisible by ?
Problem 12
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are and . Into each cone is dropped a spherical marble of radius , which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
Problem 13
What is the volume of tetrahedron with edge lengths , , , , , and ?
Problem 14
All the roots of the polynomial are positive integers, possibly repeated. What is the value of ?
Problem 15
Values for and are to be selected from without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves and intersect? (The order in which the curves are listed does not matter; for example, the choices is considered the same as the choices )
Problem 16
In the following list of numbers, the integer appears times in the list for .What is the median of the numbers in this list?
Problem 17
Trapezoid has , , and . Let be the intersection of the diagonals and , and let be the midpoint of . Given that , the length can be written in the form , where and are positive integers and is not divisible by the square of any prime. What is ?
Problem 18
Let be a function defined on the set of positive rational numbers with the property that for all positive rational numbers and . Furthermore, suppose that also has the property that for every prime number . For which of the following numbers is ?
Problem 19
The area of the region bounded by the graph of is , where and are integers. What is ?
Problem 20
In how many ways can the sequence be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
Problem 21
Let be an equiangular hexagon. The lines and determine a triangle with area , and the lines and determine a triangle with area . The perimeter of hexagon can be expressed as , where and are positive integers and is not divisible by the square of any prime. What is ?
Problem 22
Hiram's algebra notes are pages long and are printed on sheets of paper; the first sheet contains pages and , the second sheet contains pages and , and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly . How many sheets were borrowed?
Problem 23
Frieda the frog begins a sequence of hops on a grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?
Problem 24
The interior of a quadrilateral is bounded by the graphs of and , where is a positive real number. What is the area of this region in terms of , valid for all ?
Problem 25
How many ways are there to place indistinguishable red chips, indistinguishable blue chips, and indistinguishable green chips in the squares of a grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?
See also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2020 AMC 10B |
Followed by 2021 AMC 10B | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.