hugomimihu

In triangle ABC, the angle bisector of < BAC meets line segment BC at D, such that AD = AB. Line segment AD is extended to E such that CD = CE and angles DBE = BAD. SHow that triagnle ACE is iscocles.

This right?

Don't you dare cheat.

This is AoPS Week... 4/5 writing problem. From the Intro to Geometry Book.

I know since I did it a few days back.

If you think that you can just come onto some site and just copy some solution, SORRY YOU ARE OUT OF LUCK. 

I will personally not be allowing that. Many of your classmates on crunching on a solution and you just try to cheat.

If you need help, come back and 1)tell us what you did already, and/or 2) just ask the message board.

I get it, AoPS is hard, for me kinda too. But there's no shame in asking for help when you need it.

So the only squares that don't include red are 1*1, and 2*2. 

There are 25-1 such 1*1, and... 12 2*2s.

remember they overlap!

24+12 = 36.

RECTANGLES.

Remember, a square is a rectangle but a rectangle isn't a square.

Well we already have the 36 squares. Onwards!

There can be... 1*2, 1*3, 1*4, 1*5, 2*3, 2*4, 2*5. Again, these can overlap.

There *should* be 36 1*2.

Do the rest yourself and if you don't understand anything ask!

:)

Haven't done these in some time!

So. 

Bullets 2 and 3 tell us that xz = y, and yz = x.

xz = y

yz = x

Add the two equations gives

xz + yz = x+y 

= z(x + y)

z = (x+y)/(x+y) = 1!

Ayyyy!

xy - 1 = 15

xy = 16

16*1 = 16

If you don't understand anything feel free to ask.

Of course idk if this is the right answer but i'm reasonably sure that this is.

Thank you! 

But TBH, I think yours might be a bit better, but both work, and the other can be used as a different strategy to check:

essentially we did the same thing, you just subtracted the unshaded, I calculated proportion of unshaded and subtracted that from 1.

well we know that FED+ DCB count for 50% of the total area, since they each are 25%. Further, the ABF triangle is 1/8 of the total, so we have 5/8 unshaded. 640/8 = 80.

This is because if we put points...

Let's say point X is Midpoint ED, Y is midpoint CD

BCD is one half BCDX, and BCDX is one half whole rect, so 1/4

Same applies to FED.

If we put a point Z at midpoint of FY and BX, AFB is one half ABFZ, which is 1/4 whole rect, so 1/8.

3*80 = 240

if you don't understand anything feel free to ask.

ok wow thanks

didn't know that

time to add that to the list of "web2.0cac trivia"!

I actually don't know. I've had like 2 or 3 of my answers voted as "best answer". I suppose the moderator does it, since in the questions i asked, I was never able to select one, and two, many guest questions have "best answer" and usually (no offense) guests don't have many features if they don't sign up.

Of course if there's a best answer here without you picking, you can eliminate one choice :)

Since dividing by 10 gives R3, the number must end in a 3. Since remainder of /9 is 1, the multiple must be (3-1).

8*9 = 72.

72+1 = 73, which satisfies bullets 1 & 2.

73/11 = 6R7.

Answer's 7.

If you don't understand anything feel free to ask.

A 1 step stair case requirest 4, 2 step 10, so we need 6 more. 2-3 step needs 8 more,so following the pattern we get 10 toothpicsk from 3-4, 12 from 4-5.

10+12 = 22.

If you don't understand anything feel free to ask.