Can you give me the hardest math question?

Question

10 points for the first person who give me the question I can't solve. But it must have an answer.

Answer 1

Prove that:

pi^2 / 6 = 1/(1^2) + 1/(2^2) + 1/(3^2) + ...

Answer 2

When is 20/1 = 19?
When does 11+4 = 3?

Answer 3

Here's one I've been playing with for many years now:
Let n be a positive integer(in base 10).
Suppose n satisfies the following conditions:
1). n is odd
2). n is a square
3). the only digits of n are 0 and 1.
Find the first such n or prove that no such n exists.
BTW: This problem is said to have originated in a
middle-school math book. Are you smarter
than a 7th grader? (LOL).

Answer 4

Prove Cauchy's theorem for complex analysis. This says that the integral of any function around any closed path in the complex plane that does not contain a singularity of the function inside it, is zero.

By the way, the solution to ad_guy_1's problem is easy.
The line after x(x + y - z) = -y(x + y - z) should be
x = -y or x + y - z = 0 which leads to x + y = z which is what he started with anyway.

Answer 5

Fermat's (died 1655) Last Theorem states that

x^n + y^n = z^n

has no non-zero integer solutions for x, y and z when n > 2.

The proof escaped some of the best mathematicians for almost 340 years. Andrew Wiles finally solved it in 1994 and the proof was accepted in 1995.

This is my definition of "hard".

Answer 6

Determine the amount of fuel needed to travel at light speed consistantly for three years in a space craft with a total weight of 3000 pounds including passengers and equipment, but NOT including fuel.

Answer 7

Ok, here goes:

Let's assume x+y = z (where z does not equal zero)

Now lets multiply both sides by (x + y),

then, (x+y)^2 = z(x + y)

then, x^2 + y^2 + 2xy = zx + yz

then, x^2 + xy - zx = - xy - y^2 + yz

then, x(x + y - z) = -y(x + y - z)

that means, x = -y

or x + y = 0

Now, we had assumed that x + y = z and z does not equal zero. Then howcome, x + y became zero? Tell me the answer to that.

Answer 8

solve the differential equation
(x^2+1)y'+2xy=cosx+sinx, x>0
read: [(x squared) plus 1] times y prime plus 2xy equals cosine x plus sine x; x is greater than 0

Answer 9

Give an exact formula for cos(2pi/5).

Answer 10

why can't you connect all 5 points in a plane without crossing a line?

you can easily connect all 3 points. (like in a triangle)
for 4 points you can also connect (all of them) without crossing lines:
1*******2
*-*-------*
*---*-----*
*-----*---*
*-------*-*
3*******4
(with 3 and 2 connected from outside the square)