Sir-Emo-Chappington

a) A square of a prime number,

Reason: primes are the only numbers to have 2 divisors (1 and itself).

To get an odd-number of primes, the value must be a square number (Such as 25 has the divisors 5*5. Since these are both just "5" this is only going to provide 1 extra divisor, giving an odd number of divisors)

So, the only way to remain as square number which is indivisible by anything besides it's square root (and 1 and itself, of course), it's got to be a product of primes.

For the lowest possible on a single test to still result in an average value that is particularly high, then we give the other tests the highest possible value and see what we can get:

89 is the highest possible value, thus:

82 = (4*89 + fourth test) / 5

82 = (356 + fourth test) / 5

410 = 356 + fourth test

fourth test = 410 - 356

fourth test = 54

x = 29 + (18*y)

x * y = 6354

First we can insert either y or x into the other equation. For this I shall insert y's equivalent.

x * y = 6354

∴ y = ⁶³⁵⁴/_x

Insert that into the below equation like so:

x = 29 + (18*y)

∴ x = 29 + (18 * ⁶³⁵⁴/_x)

Now let's simplify and re-arrange:

x = 29 + (¹¹⁴³⁷²/_x)

Since we don't like that x being on the bottom of the fraction, let's multiply it by "x". of course everything else must be to keep this equation valid:

[x] * x = [29 + (¹¹⁴³⁷²/_x)] * x

x² = 29x + 114372

0 = -x² + 29x + 114372

We have now gotten ourselves a quadratic equation, so use the "quadratic forumula" to find all possible values of "x":

x = (-29 ± sqrt(29² - 4*-1*114372)) / (2 * -1)

x = (-29 ± sqrt(841 + 457488)) / -2

x = (-29 ± sqrt(458329)) / -2

x = (-29 ± 677) / -2

The two results are...

x = (-29 + 677) / -2 = -324

x = 353

Using these two values of x, let's find their associated values of y:

y = ⁶³⁵⁴/_x

y = ⁶³⁵⁴/_-324 = -19.61 _{[1 recurring]}

y = ⁶³⁵⁴/₃₅₃ = 18

So the two results are:

x = -324 and y = -19.61 _{[1 recurring]}

x = 353 and y = 18

2^x = 2²⁰ - 2¹⁹

Another way of reading this is:

2^x = 2²⁰ - 2^20-1

Which equates to:

2^x = 2²⁰ - (2²⁰ / 2)

Now, we get the situation of "x - (x/2) = (x/2)". This means 2^x is a half of 2²⁰, which is another way of saying:

2^x = 2¹⁹

x = 19

66666⁴ / 22222⁴

= (66666 / 22222)⁴

= (3)⁴

= 9²

= 81

Saving = old cost - new cost

Saving = ($4 * ¹⁰⁰⁰/₃₀ per month for a year) - ($4 * ¹⁰⁰⁰/₅₀ per month for a year)

Saving = (⁴⁰⁰⁰/₃₀ * 12) - (⁴⁰⁰⁰/₅₀ * 12)

Saving = (1600) - (960)

Saving = $640 a year

I hope I understood the question correctly:

We have 2 wins, and 8 loses. They then win x amount more and end on equal loses and wins

So:

2 + x = 8

x = 6

10 meets + 6 = 16 total meets.

I'm not sure if you meant "(pi * y)^3" or "pi * (y^3)".

If you meant "x = (π*y)³":

x = (π*y)³

y = sqrt(78)

you can replace then "y" in x's equation with sqrt(78) and work from there:

x = (π * sqrt(78))³

Let's expand this out to make it easier to work with.

x = π³ * sqrt(78)³

Seeing as "sqrt(78)" is being raised to a power above 2, it's possible to simplify it further.

x = π³ * sqrt(78)² * sqrt(78)

x = π³ * 78 * sqrt(78)

There's not much left to do other than to multiply it all together.

x = 31.006 * 78 * 8.832

x = 21360

If you meant "x = n * (y³)":

x = π * (y³)

y = sqrt(78)

Again, let's just insert y into x's equation.

x = π * (sqrt(78))³

Again, we can simplify that square root down at least a little.

x = π * 78 * sqrt(78)

And now let's finish the equation.

x = 3.142 * 78 * 8.832

x = 2164.5

Why would you ever need such an absurdly huge number? That's more than a Googolplex to the Googolplex!

(It's also literally impossible to write out the number because of the inherent nature of the scale of a Googolplex)

Depends: what exactly did you take away 9 of?

either: 21 - 9 = 12

or: 21 - 0 = 21