mathleteig

Wait, isn't it -4.

Ok, so this nimrod of a student was supposed to calculate 2 times x cubed. Instead, he computed 2x cubed. (2x)³ = 8x³

Now, we know that 2x^3 is 384 greater than the correct answer which is 8x^3.

8x^3 + 384 = 2x^3

-6x^3 = 384

x^3 = -64

x = -4

If needed, the diagram looks like this...

I hope you enjoyed getting schooled by a seventh grader! I'm just kidding. I wish you the best of luck for your MATHCOUNTS entry exam.

Go back to that Venn diagram.

It would have a place for FICTION, a place for NONFICTION, and then an intersection. We have to subtract eight from both because 8 people like both. If they like both, that means they are included in the place where people like fiction, in the place where people like nonfiction, and the intersection. We don't want this to be counted three times. Instead, we subtract 8 from both main quantities and just leave the thing that is common. If you haven't already, you should look at the basics of sets. It kind of gets into this topic once you learn the main idea.

A Venn diagram is a good way to visualize these kind of problems, but if you get more familiar with the concept and how you can manipulate the numbers, it will be way easier than drawing out a Venn diagram. Out of all 100 people, 8 people liked both non fiction and fiction. Then, it tells us three times as many people like fiction over nonfiction. Consider nonfiction to be x.

3x like fiction and x and non fiction. We also have to look at the 8 people who like both. They are included in the people who like fiction and non fiction. In order to get an accurate response... you have to do the subtract the 8 out of the total.

FICTION - 8 + NONFICTION - 8 + 8 = TOTAL

3x - 8 + x - 8 + 8 = 100

4x - 8 = 100

4x = 108

x = 27

Since we said x was non-fiction, 27 people reported liking non-fiction.

The radius of the circle is the circumradius. It is called a circumradius because the triangle insribed in the circle has a circumcenter O and O is also the center of the circle

The diagram looks a litte bit like this...

There is no methodical reasoning behind five-digit perfect square palindromes. It's a matter of squaring all the numbers from 100 to 316.

(317² = 100489)

101² = 10201

111² = 12321

121² = 14621

202² = 40804

212² = 44944

264²  = 69696

307² = 94249

For four-digit perfect square plaindromes... there are none!

So, your answer would be the 7 five-digit perfect square palindromes that we stated above.

If a, b, or c = 0, then there is 15 reversable ordered pairs for any of the two other variables adding up to 30. 

( a, b ) = 30

( b, c ) = 30

( a, c ) = 30 

15 * 3 = 45. Since the ordered pairs are reversable between two variables, 45 * 2 = 90 

Similarly, if a, b, or c = 1, then there is 14 reversable ordered pairs for any of the two variables adding up to 29. 

14 * 3 * 2 = 84 

...

I advise to continue on with this method.