Search results

A harder question along those lines, prove that the sum: 1+1/2+1/3+...+1/n is not an integer for any n>1. Also, I posted a question on the previous page.

Suppose the sum is equal to some integer N. Multiply both sides by p1p2...pn and you will get an equation, the right side of which is divisible by every prime in the set, the left side of which isn't. Contradiction.

All three unknowns are positive integers, that is the only restriction.

Here's one I really liked in high school: Find all solutions in positive integers to: x^2+y^2+1=xyz.

Only nontrivial solns are the pairs (4,2), (2,4). Proof: $For $x\geq 3>e$, and $k$ any positive integer we have:\\$(x+k)^x=x^x(1+k/x)^x<x^xe^k<x^{x+k} so we cannot have equality if x and y are not equal. Here I have assumed a commonly known limit expression for e (and the fact that this...

I got: \frac{11\pi}{24\sqrt{3}}\approx 83.1\%

Not sure if I agree with this. Whilst there is some overlap in areas, university does things properly from scratch, something that I don't think is possible with the time constraints of one year and the level of understanding of the average high school teacher. I think that universities should...

Yeah...they seem very easy. If you want a taste for good problemsolving mathematics try some olympiad stuff, it is far more challenging.

And yes the initial beam is fired FROM a focus. Or equivalently, from the boundary directly towards a focus.

Showing the reflective angles decrease doesn't necessarily imply they converge to zero, although that approach does seem fruitful. I was thinking more along the lines of finding a recursive relation involving the parameters describing the points where the ray meets the boundary. If the...

Pretty sure this integral can only be expressed in terms of hypergeometric functions...

Re: 2012 HSC MX2 Marathon If you have trouble spotting any clever substitution to kill such integrals, always remember the brute-force methods. Partial fractions works fine here, as it does for the previous integral.

Haha I wish. No this is a well known fact, and there are definitely books containing a proof. It's only loosely related to my actual research.

Here is a question that is vaguely related to my research. I am not sure how difficult it is to prove, but some of you might find it interesting. Consider an elliptical region bounded by a mirror. A light ray is emitted from one focus of this ellipse in any direction, and proceeds to bounce off...

Nice way of looking at it! Of course the direct proof of the recursive relation is just summing the equation (x^k)P(x)=0 over the roots of P, where k is a natural number. Would rep but need to share it apparently...

If x,y,z are complex numbers such that: x+y+z=0, prove that: \frac{x^{11}+y^{11}+z^{11}}{11}=\frac{x^3+y^3+z^3}{3}\cdot\frac{x^8+y^8+z^8}{2}-\frac{(x^3+y^3+z^3)^3}{9}\cdot\frac{x^2+y^2+z^2}{2}.

Cool, will come back later and look through my working more carefully.

What was wrong?

Mine came out numerically as approximately 0.0790, although I am not entirely confident in it.

I get the feeling it should always exist, the area will just get smaller and smaller. Have something I've got to do right now but will look at this again later. I'm not sure if my method is convenient to adapt to n-gons. Okay, I have worked out a different method that should work for n-gons...