Difference between revisions of "2022 AMC 12B Problems"
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==Problem 9 == | ==Problem 9 == | ||
+ | The sequence <math>a_0,a_1,a_2,\cdots</math> is a strictly increasing arithmetic sequence of positive integers such that <cmath>2^{a_7}=2^{27} \cdot a_7.</cmath> What is the minimum possible value of <math>a_2</math>? | ||
− | + | <math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 16 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 22</math> | |
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− | <math>\textbf{(A)}8 \qquad \textbf{(B)}12 \qquad \textbf{(C)}16 \qquad \textbf{(D)}17 \qquad \textbf{(E)}22</math> | ||
[[2022 AMC 12B Problems/Problem 9|Solution]] | [[2022 AMC 12B Problems/Problem 9|Solution]] |
Revision as of 21:18, 17 November 2022
2022 AMC 12B (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 |
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
Define to be
for all real numbers
and
. What is the value of
Problem 2
In rhombus , point
lies on segment
so that
,
, and
. What is the area of
? (Note: The figure is not drawn to scale.)
Problem 3
How many of the first ten numbers of the sequence ,
,
, ... are prime numbers?
Problem 4
For how many values of the constant will the polynomial
have two distinct integer roots?
Problem 5
The point is rotated
counterclockwise about the point
. What are the coordinates of its new position?
Problem 6
Consider the following sets of
elements each:
How many of these sets contain exactly two multiples of
?
Problem 7
Camila writes down five positive integers. The unique mode of these integers is greater than their median, and the median is
greater than their arithmetic mean. What is the least possible value for the mode?
Problem 8
What is the graph of in the coordinate plane?
Problem 9
The sequence is a strictly increasing arithmetic sequence of positive integers such that
What is the minimum possible value of
?
Problem 10
Regular hexagon has side length
. Let
be the midpoint of
, and let
be the midpoint of
. What is the perimeter of
?
Problem 11
Let , where
. What is
?
Problem 12
XXX
Problem 13
XXX
Problem 14
XXX
Problem 15
XXX
Problem 16
XXX
Problem 17
How many arrays whose entries are 0s and 1s are there such that the row sums (the sum of the entries in each row) are 1, 2, 3, and 4, in some order, and the column sums (the sum of the entries in each column) are also 1, 2, 3, and 4, in some order? For example, the array
satisfies the condition.
Problem 18
XXX
Problem 19
XXX
Problem 20
XXX
Problem 21
XXX
Problem 22
XXX
Problem 23
Let be a sequence of numbers, where each
is either
or
. For each positive integer
, define
Suppose for all
. What is the value of the sum
Problem 24
XXX
Problem 25
XXX