Difference between revisions of "2021 AMC 10A Problems"
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==Problem 22== | ==Problem 22== | ||
+ | Hiram's algebra notes are <math>50</math> pages long and are printed on <math>25</math> sheets of paper; the first sheet contains pages <math>1</math> and <math>2</math>, the second sheet contains pages <math>3</math> and <math>4</math>, 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 <math>19</math>. How many sheets were borrowed? | ||
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+ | <math>\textbf{(A)} ~10\qquad\textbf{(B)} ~13\qquad\textbf{(C)} ~15\qquad\textbf{(D)} ~17\qquad\textbf{(E)} ~20</math> | ||
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==Problem 23== | ==Problem 23== | ||
==Problem 24== | ==Problem 24== | ||
==Problem 25== | ==Problem 25== |
Revision as of 15:06, 11 February 2021
2021 AMC 10A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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 |
February 4, 2021, is when the AMC 10A starts.
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
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 of 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 knows that:
All of his happy snakes can add
None of his purple snakes can subtract
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 integer
?
Problem 9
What is the least possible value of for real numbers
and
?
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
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
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
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?