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