Difference between revisions of "2021 Fall AMC 12A Problems"
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==Problem 19== | ==Problem 19== | ||
+ | Let <math>x</math> be the least real number greater than <math>1</math> such that sin<math>(x)</math> = sin<math>(x^2)</math>, where the arguments are in degrees. What is <math>x</math> rounded up to the closest integer? | ||
+ | <math>\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 14 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 20</math> | ||
[[2021 Fall AMC 12A Problems/Problem 19|Solution]] | [[2021 Fall AMC 12A Problems/Problem 19|Solution]] |
Revision as of 19:05, 23 November 2021
2021 Fall AMC 12A (Answer Key) Printable versions: • Fall AoPS Resources • Fall 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 |
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
Menkara has a index card. If she shortens the length of one side of this card by
inch, the card would have area
square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by
inch?
Problem 3
Mr. Lopez has a choice of two routes to get to work. Route A is miles long, and his average speed along this route is
miles per hour. Route B is
miles long, and his average speed along this route is
miles per hour, except for a
-mile stretch in a school zone where his average speed is
miles per hour. By how many minutes is Route B quicker than Route A?
Problem 4
The six-digit number is prime for only one digit
What is
Problem 5
Elmer the emu takes equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in
equal leaps. The telephone poles are evenly spaced, and the
st pole along this road is exactly one mile (
feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
Problem 6
As shown in the figure below, point lies on the opposite half-plane determined by line
from point
so that
. Point
lies on
so that
, and
is a square. What is the degree measure of
?
Problem 7
A school has students and
teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are
and
. Let
be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let
be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is
?
Problem 8
Let be the least common multiple of all the integers
through
inclusive. Let
be the least common multiple of
and
What is the value of
Problem 9
A right rectangular prism whose surface area and volume are numerically equal has edge lengths and
What is
Problem 10
The base-nine representation of the number is
What is the remainder when
is divided by
Problem 11
Consider two concentric circles of radius and
The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle?
Problem 12
What is the number of terms with rational coefficients among the terms in the expansion of
Problem 13
Each of balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other
balls?
Problem 14
Problem 15
Problem 16
Problem 17
For how many ordered pairs of positive integers does neither
nor
have two distinct real solutions?
Problem 18
Each of balls is tossed independently and at random into one of
bins. Let
be the probability that some bin ends up with
balls, another with
balls, and the other three with
balls each. Let
be the probability that every bin ends up with
balls. What is
?
Problem 19
Let be the least real number greater than
such that sin
= sin
, where the arguments are in degrees. What is
rounded up to the closest integer?
Problem 20
For each positive integer , let
be twice the number of positive integer divisors of
, and for
, let
. For how many values of
is
Problem 21
Problem 22
Problem 23
A quadratic polynomial with real coefficients and leading coefficient is called
if the equation
is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial
for which the sum of the roots is maximized. What is
?
Problem 24
Problem 25
See also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2021 AMC 12B Problems |
Followed by 2021 Fall AMC 12B 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.