2016 AMC 12A
Problem 1
What is the value of
?
Solution 1
Factoring out from the numerator and cancelling out from the numerator and the denominator, we have
Solution 2
We can use subtraction of fractions to get
Solution 3
Factoring out
gives
Problem 2
For what value of
does
?
Solution 1
We can rewrite as :
. Solving the equation
Since the bases are equal, we can set the exponents equal, giving us gives us
Solution 2
We can rewrite this expression as to to
, which can be simplified
, and that can be further simplified
. This leads to
. Solving this linear equation
yields
Problem 3
For every dollar Ben spent on bagels, David spent
cents less. Ben paid
more than David.
How much did they spend in the bagel store together?
Solution
If Ben paid
more than David, then he paid
. Thus, David
paid
, and they spent .
Problem 4
The remainder can be defined for all real numbers
and with
by
where
is the value of
?
denotes the greatest integer less than or equal to
. What
Solution
The value, by definition, is
Problem 5
A rectangular box has integer side lengths in the ratio volume of the box?
. Which of the following could be the
Solution
Let the smallest side length be then
. Then the volume is
.If
,
Problem 6
Ximena lists the whole numbers through much larger is Ximena's sum than Emilio's?
once. Emilio copies Ximena's numbers, replacing each
occurrence of the digit by the digit . Ximena adds her numbers and Emilio adds his numbers. How
Solution
For every tens digit 2, we subtract 10, and for every units digit 2, we subtract 1. Because 2 appears 10 times as a tens digit and 2 appears 3 times as a units digit, the answer is
Problem 7