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A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are <math>57</math>, <math>60</math>, and <math>91</math>. What is the fourth term of this sequence? | A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are <math>57</math>, <math>60</math>, and <math>91</math>. What is the fourth term of this sequence? | ||
− | <math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } | + | <math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } 206</math> |
[[2022 AMC 10A Problems/Problem 20|Solution]] | [[2022 AMC 10A Problems/Problem 20|Solution]] |
Revision as of 22:47, 11 November 2022
2022 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 |
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
Mike cycled laps in
minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first
minutes?
Problem 3
The sum of three numbers is The first number is
times the third number, and the third number is
less than the second number. What is the absolute value of the difference between the first and second numbers?
Problem 4
In some countries, automobile fuel efficiency is measured in liters per kilometers while other countries use miles per gallon. Suppose that 1 kilometer equals
miles, and
gallon equals
liters. Which of the following gives the fuel efficiency in liters per
kilometers for a car that gets
miles per gallon?
Problem 5
Square has side length
. Points
,
,
, and
each lie on a side of
such that
is an equilateral convex hexagon with side length
. What is
?
Problem 6
Which expression is equal to for
Problem 7
The least common multiple of a positive divisor and
is
, and the greatest common divisor of
and
is
. What is the sum of the digits of
?
Problem 8
A data set consists of not distinct) positive integers:
,
,
,
,
, and
. The
average (arithmetic mean) of the
numbers equals a value in the data set. What is
the sum of all positive values of
?
Problem 9
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy] size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); [/asy]
Problem 10
XXX
Problem 11
Ted mistakenly wrote as
What is the sum of all real numbers
for which these two expressions have the same value?
Problem 12
On Halloween children walked into the principal's office asking for candy. They
can be classified into three types: Some always lie; some always tell the truth; and
some alternately lie and tell the truth. The alternaters arbitrarily choose their first
response, either a lie or the truth, but each subsequent statement has the opposite
truth value from its predecessor. The principal asked everyone the same three
questions in this order.
"Are you a truth-teller?" The principal gave a piece of candy to each of the
children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the
children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the children who
answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
Problem 13
XXX
Problem 14
XXX
Problem 15
XXX
Problem 16
XXX
Problem 17
How many three-digit positive integers
are there whose nonzero digits
,
, and
satisfy
(The bar indicates repetition, thus in the infinite repeating decimal
Problem 18
XXX
Problem 19
XXX
Problem 20
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are ,
, and
. What is the fourth term of this sequence?
Problem 21
A bowl is formed by attaching four regular hexagons of side to a square of side
. The edges of the adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?
Problem 22
XXX
Problem 23
XXX
Problem 24
XXX
Problem 25
XXX
See also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2021 Fall AMC 10B Problems |
Followed by 2022 AMC 10B 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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.