CentsLord

(answer adapted from Balaji Reddy on Quora)

First of all, let's see how many one-row rectangles there are.

1 x 1 rectangles = (9–1)*(9–1) = 64

1 x 2 rectangles = (9–1)*(9–2) = 56

1 x 3 rectangles = (9–1)*(9–3) = 48

1 x 4 rectangles = 5 * 8 = 40

1 x 5 rectangles = 4 * 8 = 32

1 x 6 rectangles = 3 * 8 = 24

1 x 7 rectangles = 2 * 8 = 16

1 x 8 rectangles = 1 * 8 = 08

Total = 64+56+48+40+32+24+16+8 = 288 1-row rectangles!

There's a pattern here :D

Total one row rectangles = 8*1+8*2+…+8*8

= 8 * (1+2+…+8)

= 8 * (8*9/2) Since,n(n+1)/2 = sum of all numbers below n

=8 * 36

Similarly, Two row rectangles = 7 * (1+2+…+8)

=7 * 36

So all rows rectangles = 8*36+7*36+…+1*36

= 36 * (1+2+…+8)

= 36 * 36

It's \(\fbox{1296}\) :)

BTW, this also counts all the squares!

Epic!

Sorry! Well, I tried :(

The total ways to make sequences of 4 letters using the letters P, R, O, B, L, E, M, using each letter only once, without restrictions, is 7 * 6 * 5 * 4 = 840.

Let's use complementary counting for this problem (So, we find the situations we DON'T want and subtract them from the total)! :D

Ways that a sequence begins with L = 1 * 6 * 5 * 4 = 120

Ways that a sequence ends with P = 6 * 5 * 4 * 1 = 120

Ways that a sequence begins with L and ends with a P = 1 * 5 * 4 * 1 = 20

120 + 120 - 20 = 220

There are 220 ways which we DON'T want!

So, since the total is 840, we subtract 840 from 220 to get \(\fbox{620}\) :D

EDIT: Sorry I misread the problem! This answer is false.

A Simple question! Give each child 5 /4 of a muffin!

Lol, clearly you're from AoPS. I can see your Avatar was created by a user... I forgot their name but they've made many avatars xD

So we know that \(\dfrac{3+x}{1+x} = k\), and \(\dfrac{8+x}{3+x} = k\)

8 + x = 3k + kx

3 + x = k + kx

Subtracting the second equation from the first one, we get 5 = 2k 

This means k = 5/2 = 2.5 :D

Putting that into the second equation, we get:

3 + x = 2.5 + 2.5x

0.5 = 1.5x

x = 0.5 / 1.5 = 1/3

So, your answer is \(\fbox{$\dfrac{1}{3}$}\) :D

\(4 \cdot 6 = \fbox{$24 \text{ hours.}$} \text{ Thank you, ryanthemathboi, very cool xD}\)

You can't get downvoted by guests. 

xD I think I might have figured it out!

There's just BDAC and CADB

So it's just 2 ways, I think?