Difference between revisions of "2012 AMC 12A Problems"
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== Problem 3 == | == Problem 3 == | ||
+ | A box <math>2</math> centimeters high, <math>3</math> centimeters wide, and <math>5</math> centimeters long can hold <math>40</math> grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold <math>n</math> grams of clay. What is <math>n</math>? | ||
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+ | <math>\textbf{(A)}\ 120\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 200\qquad\textbf{(D)}\ 240\qquad\textbf{(E)}\ 280</math> | ||
== Problem 4 == | == Problem 4 == |
Revision as of 16:42, 11 February 2012
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
A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether?
Problem 2
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?
Problem 3
A box centimeters high,
centimeters wide, and
centimeters long can hold
grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold
grams of clay. What is
?
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Let ,
, and
be positive integers with
such that
and
What is ?
Problem 22
Problem 23
Let be the square one of whose diagonals has endpoints
and
. A point
is chosen uniformly at random over all pairs of real numbers
and
such that
and
. Let
be a translated copy of
centered at
. What is the probability that the square region determined by
contains exactly two points with integer coefficients in its interior?
Problem 24
Let be the sequence of real numbers defined by
, and in general,
\[a_k=\left\{\array{c}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if k is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if k is even.}\] (Error compiling LaTeX. Unknown error_msg)
Rearranging the numbers in the sequence in decreasing order produces a new sequence
. What is the sum of all integers
,
, such that
$\textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\2012$ (Error compiling LaTeX. Unknown error_msg)
Problem 25
Let where
denotes the fractional part of
. The number
is the smallest positive integer such that the equation
has at least
real solutions. What is
? Note: the fractional part of
is a real number
such that
and
is an integer.
$\textbf{(A)}\ 30\qquad\textbf{(B)}\ 31\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 62\qquad\textbf{(E)}\64$ (Error compiling LaTeX. Unknown error_msg)