Forgo

If the average of 10 numbers is 10, the sum of the numbers is \(10*10 \) or 100. If we remove a number, we get the following equation:

                                                                                                          \(\frac{100-x}{9} = 9\)

Now, we solve for \(x\):

\(\frac{100-x}{9} = 9\)

 \(900 - 9x = 81\)

We divide one side by 9 which results to:

\(100 - x = 81 \)

\(100 - 81 = 19 \)

We get the final result which is \(19=x\).

All the work he's done for the past 6 years... All gone

:((((((((((((((((((((((

We subtract the area of the bigger square from the area of the smaller square.

(x+7)(x+5) = x^2+12x+35

The larger rectangle has an expanded area of 2x^2 -7x + 6.

We subtract (2x^2 - 7x + 6) - (x^2+12x+35)

This gives us x^2 - 19x - 29

Ummmmmmmmmmmmmmmmmmm.........

Another Aops Question...

Hint:

\(\binom{n}{k}=\binom{n}{n-k}\)

Stop posting Art of problem solving questions!

The amount of work to get the answer is dazzling.

We get the equations:

25q + 10d = 265

q > d.

Divide by 5 for the first equation yields us 5q + 2d = 53.

After trial and error the value of q = 9, d = 4

To check: 

25 x 9 = 15 x 15 = 225.

4 x 10 = 40.

225+40=265...

Yeah, I think itś 2^6.

There are 2 possibilities for the first

There are 2 for the second

There are 2 for the third. This goes on...

Your not wrong when you say this is hard...

I'll take a shot at it...

I got the values x = 2

and x = -1

They seem to work...

I first factored 4x^2 - 1 to get (2x + 1)(2x - 1). When we add fractions we multiply the denominators to get the common denominator.

When I factored 4x^2 - 1 I got (2x + 1)(2x - 1) and that is the same denominator when you multiply the other 2 denominators together.

We get 4x/(2x + 1)(2x - 1)

We multiply 1 by (2x + 1)(2x - 1) so the denominator cancels out.

We get 4x + 7 = (2x + 1)(2x - 1)

We expand (2x + 1)(2x - 1) to get 4x^2 - 1. 

4x + 7  = 4x^2 - 1

We subract 7 on both sides to receive 4x^2 - 8 = 4x

Then, we subtract 4x from both sides to get 4x^2 - 8 - 4x = 0

We can use the quadratic formula to find the solution from here on out.

a = 4, b = -4, c = -8

\(x = {-4 + \sqrt{(-4)^2-4(4(-8)} \over 2(4)}\)

Saving you time, this gives us 2 and -1.

If this question was assigned to you your learning about the pigen hole principle.

You have to look at the worst case to ensure that you will get it.

If your really unlucky, you might get 10 yellow socks in a row and 13 red socks (soxs) in a row.

Then, it is certain that you will get an either GREEN SOCK, or YELLOW SOCK so we add 1.

10 + 13 + 1 = 24

Red Sox pun not intended but definitely appreciated...

(-1)/2, -(1/2), -1(-2) is the same thing as -1/2 as a whole. 

They are all the same thing...

Probabiliy = Desired Outcomes/All possible outcomes

1. All possible outcomes = 5

   Desired Outcomes = 1

So, the probability of losing a turn on the first spinner is 1/5

2. There is a 2/5 chance that you will either get a "Move Up" or "Move Right".

We cube 2/5 b/c we can't have use any other moves.

2/5 x 2/5 x 2/5 = 8/125

You try three yourself wih the information I have given you

A-----------------------B

0 ---------------------- 180

The trip from A to B took him 3 hrs

The trip from B to A took 2hrs.

\(\frac52\) = 2.5

BTW, That's incorrect.

Label one vertex of the heptagon as any variable you choose.

The O's can't be next to the point N.

There are \(\binom42 = 6\) ways to choose the 2 spots. Since there are 4! ways to label the other vertices, we multiply 4! = 24 by 6 to get 144.

DO NOT COPY THIS SOLUTION WORD FOR WORD.

I know this is an AoPs problem but I also rlly needed help so.. Yeah.

Make sure to understand method used to solve this problem

Ok, This is very obsecure but there is a such thing as a cube difference rule.

This falls into the cube difference rule.

If we were to expand (x+y)^3 we would get: 

x^3+3x^2y+3xy^2+y^3

You can see the similarities...

Anyway, the answer is (x - 2)^3

Also, you can use the binomial theorem

Ok, 

Let's have Jane's age as "j" and Andrew's age as "a".

j = 4a

j+1 = 3(a+1)

We convert j into 4a

4a + 1 = 3a + 3

4a = 3a + 2

a = 2

If a = 2

j = 8

Andrew is 2 years old

Jane is 8 yrs old

(As of now)

Well, 0 is an imaginary number so.............

\(0i\)

:)