Challenging Math Questions

[bomaha]

Ok so I want to make a topic containing challenging math questions (kind of math riddles) that stimulate thinking.

So the rule is:

Once someone put a question no other person can ask another question unless the first question is answered with explanation.

And please don't post IB questions. Please don't get the answers from the internet. Try to do it yourself. If you can't, leave the answer room for other students.

I hope I was clear.

So I will begin:

or if you can't see the figure, here is a link: http://i.imgur.com/VlbOC.png

In the figure above, B is the midpoint of AC and the center of the green circle. All other labeled points are also the centers of circles. If the green area is 9π, what is the area of the circle with center E?

Edited November 12, 2011 by bomaha

[bomaha]

Nice try nametaken but your answer is wrong

[ILM]

So this is my answer, i called circles by their center:

Okay i have a very simple question, depending in your speed of answering it i will post other questions.

The question says

Proof that:

b²+a² >= 2ab

[bomaha]

Correct

Ok My answer is:

For any a and b the following equation is true:

Let's say that (a-b)^2 >= 0

a^2 - 2ab + b^2 >= 0

a^2 + b^2 >= 2ab

My question is:

Let P be a point inside a square S so that the distances from P to the four vertices, in order, are 7, 35, 49, and x. What is x?

Edited January 6, 2012 by King Glau

[dessskris]

let's go for the more challenging ones.

which one is greater:

or

?

[nametaken]

You could round e to 3, and pi to 4. So you're rounding up.

e^pi : 3^4=81

pi^e:4^3= 64

Therefore e^pi is greater than pi^e.

[bomaha]

I kind of asked a question before you Desy but what's pi? Do you mean (3.1415...) or is it different variables? Can I use a calculator or do you want the reason without it?

Edited November 12, 2011 by bomaha

[dessskris]

your proof is inadequate nametaken

bomaha yes I'm referring to that pi.

provide a reason without calculator, please show why is exactly greater than .

well to be fair you can start with the knowledge that but prove that it is true.

oh and I don't get why the pi is not showing up, it works just fine on TSR!

Edited November 12, 2011 by Desy Glau

[bomaha]

I want to get this thread going so (i'm not sure if my proving is right):

e^pi > pi^e

ln(e^{pi}) > ln(pi^e)

pi > e ln(pi)

pi / ln(pi) > e

take the function f(x)= x/ln(x)

Find the minimum: x = e, so

pi / ln(pi) > e / ln(e)

pi > ln (pi^e)

Raising by e:

e^pi > pi^e

My question is:

-- Let P be a point inside a square S so that the distances from P to the four vertices, in order, are 7, 35, 49, and x. What is x?

Edited November 19, 2011 by bomaha

[IBCONQUERER]

find common difference 35 - 7 = 28

49 - 35 = 14

means there is a quadratic relationship between the distances, because there seems to be some pattern between the differences in distance

so ax^2 + bx + c = d

a(1)^2 + b(1) + c= 7

a(2)^2 + b(2) + c = 35

a(3)^2 + b(3) + c= 49

now solve simultaneously

a + b + c = 7

4a + 2b + c = 35

9a + 3b + c = 49

equate (i) and (ii) and (ii) and (iii)

3a + b = 28

5a + b = 14

-2a = 14

a = -7

3(-7) + b = 28

28 + 21 = b = 49

c = 7 + 7 - 49 = -35

so -7x^2 + 49x - 35 = d

so now for the fourth distance simply plug in 4 in place of x

-7(4)^2 + 49(4) - 35 = -7(16) + 196 - 35 = -112 - 35 + 196 = -147 + 196 = 49

Am I right?