The Proof is Trivial!

Problem 494 ***

Evaluate:
$\displaystyle \int_{0}^{\infty }\frac{cos(ax)}{1+x^2}dx$ for $a\geq 0$.

Sorry if it's been asked before.

Is applied mathematics allowed on here?

Yeah. Maybe if it's some easy-ish applied maths I might be able to actually answer a question.

From IMO 2013 Q1.

Problem 492**

Prove that for any pair of positive integers $k$ and $n$, there exist $k$ positive integers $m_1,m_2,...,m_k$ (not necessarily different) such that

$1 + \dfrac{2^k - 1}{n} = \left(1 + \dfrac{1}{m_1}\right)\left(1 + \dfrac{1}{m_2}\right)...\left(1 + \dfrac{1}{m_k}\right)$

After 15 hours, I have finally solved it... Finally...
Now I can do some revision haha

Nice. Although I'm not sure if I should feel somewhat guilty for distracting you .

After 15 hours, I have finally solved it... Finally...
Now I can do some revision haha

Proof or gtfo

Fine then (warning, slightly long and inelegant)

Niiiiice. Basically by induction on the binary expansion, I suppose, but you've gone and done a constructive proof.

There are hundreds of techniques when it comes to functional equations. It is annoying that there are no books on functional eqs; I could think of 1-2 really good for olympiad problems.


.

plz can you share them
the methods and techniques
im aweful

At and below BMO2 level, substitution is the main one. Sub in some values, see if you get any good relations, see if you can get any values for some f(x). At a higher level, check for injectivity and surjectivity (you may have to look on wikipedia if you don't know what these are). Functional equations also get more algebraic as they get harder, so then a set method doesn't really apply. Someone probably has better advice than this.

do you know whether for q 5 0 counts as being in the set of ''non negative integers'' as the positive integer sign means positive lol (which doesnt include 0)/
http://www.bmoc.maths.org/home/bmo1-2002.pdf
Thank you.

do you know whether for q 5 0 counts as being in the set of ''non negative integers'' as the positive integer sign means positive lol (which doesnt include 0)/
http://www.bmoc.maths.org/home/bmo1-2002.pdf
Thank you.

0 is indeed in the set of non-negative integers, although I always thought that Z+ was the set of positive integers.

precisely why i am confuzzled by this

In the question, I guess its the set of non-negative integers then.