2010 AMC 12A Problems and Solution

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Solution 2

We can use Principle of Inclusion-Exclusion to find the final volume of the cube. There are 3 \\

volumes, as the central

cubic inches. However, we can not just sum their

cube is included in each of these three cuts. To

get the correct result, we can take the sum of the volumes of the three cuts, and subtract the volume of the central cube twice. Hence the total volume of the cuts is

.

Therefore the volume of the rest of the cube is .

Solution 3

We can visualize the final figure and see a cubic frame. We can find the volume of the figure by adding up the volumes of the edges and corners. Each edge can be seen as a

box.

box, and each corner can be seen as a

.

Solution 4

First, you can find the volume, which is 27. Now, imagine there are three prisms of dimensions 2 x 2 x 3. Now subtract the prism volumes from 27. We have -9. From here we add two times 2^3, because we over-removed (LOL). This is 16 - 9 = 7 (A).

Problem 10

The first four terms of an arithmetic sequence are , , is the

term of this sequence?

, and

. What

Solution

and

are consecutive terms, so the common difference is

.

The common difference is . The first term is and the

term is

Problem 11

The solution of the equation What is ?

can be expressed in the form

.

Solution

This problem is quickly solved with knowledge of the laws of exponents and logarithms.

Since we are looking for the base of the logarithm, our answer is .

Problem 12

In a magical swamp

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