Search found 1015 matches
- Thu Mar 05, 2015 7:58 pm
- Forum: Secondary Level
- Topic: REciprocals!!
- Replies: 3
- Views: 4055
Re: REciprocals!!
Google 'Egyptian Fractions' and you will have your ans.
- Thu Mar 05, 2015 7:55 pm
- Forum: Number Theory
- Topic: Form of the divisors
- Replies: 2
- Views: 3130
Re: Form of the divisors
@tanmoy, I suppose you were expecting a detailed ans? Ignore this post, if that's not the case. It can be proved if $p|x^2+y^2$ and $p\not | x,y$, then $p=4k+1$ for some $k$. Let $x^2+y^2\equiv 0\pmod p$ $\Longrightarrow -x^2\equiv y^2\pmod p$ $\Longrightarrow (-x^2)^{\frac {p-1}{2}}\equiv (y^2)^{\f...
- Thu Mar 05, 2015 7:41 pm
- Forum: Secondary Level
- Topic: (3n+1)^2+4n^3=m^2
- Replies: 3
- Views: 3556
Re: (3n+1)^2+4n^3=m^2
Typically these equations are called Diophantine Equations. And it's part of Number Theory. I don't know if there is any good (olympiad level) material on this topic. Try "Introduction to Diophantine Equations", perhaps? Also, in this thread, you are not required to solve that equation. It was actua...
Re: Minimize!
Let $a+b=x,b+c=y,c+a=z$. The given condition becomes $4\ge x^2+y^2+z^2\quad (1)$ \[\begin{align*}S &= \sum_{cyc} \frac{(x-y+z)(y-z+x)+4}{4x^2}\\ & = \frac 3 4 + \sum_{cyc} \frac{4-(y-z)^2}{4x^2}\\ & \ge \frac 3 4 +\sum_{cyc} \frac{x^2+2yz}{4x^2}\quad [\text{Using (1)}]\\ & \ge \frac 3 4 + \frac 3 4 ...
- Mon Feb 23, 2015 9:29 pm
- Forum: Secondary Level
- Topic: (3n+1)^2+4n^3=m^2
- Replies: 3
- Views: 3556
(3n+1)^2+4n^3=m^2
Prove that there exist infinitely many positive integer $n$s such that $(3n+1)^2+4n^3$ is a perfect square.
Hint:
Hint:
- Wed Feb 18, 2015 11:31 am
- Forum: Combinatorics
- Topic: Iran 3rd round 2013
- Replies: 4
- Views: 3851
Re: Iran 3rd round 2013
Seriously, nobody loves combi? I am really surprised not seeing any other post in this thread even after two days. Anyway, here is a hint:
- Wed Feb 11, 2015 1:22 pm
- Forum: Number Theory
- Topic: Iran TST 2011 D4 P3
- Replies: 1
- Views: 2542
Iran TST 2011 D4 P3
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that \[af(a)+bf(b)+2ab\in \square \forall (a,b)\in \mathbb N^2\]Here $\square =\{a^2:a\in \mathbb N\}$
- Fri Feb 06, 2015 9:41 am
- Forum: College / University Level
- Topic: Integrability
- Replies: 3
- Views: 10279
Re: Integrability
Well, some non-elementary functions can be integrated (Yeah, the blessings of power series. :D But this is not a general approach, as they don't always converge to the original function.) See more here: http://en.wikipedia.org/wiki/Nonelementary_integral @Rabeeb, here's the answer of your question: ...
- Fri Feb 06, 2015 9:15 am
- Forum: Secondary Level
- Topic: Practice Problem Set for National (Secondary Category)
- Replies: 2
- Views: 5170
Re: Practice Problem Set for National (Secondary Category)
Thank you for compiling and posting these. I hope you wouldn't mind if I keep them both in the website of MPMS, would you?
- Sun Feb 01, 2015 11:10 am
- Forum: Number Theory
- Topic: Cool NT
- Replies: 1
- Views: 2445
Cool NT
A function $f:\mathbb Z \to \mathbb Z$ satisfies \[f(m)+f(n)+f(f(m^2+n^2))=1\quad \forall m,n\in \mathbb Z\] Also $\exists a,b:f(a)-f(b)=3$. Prove that $\exists c,d:f(c)-f(d)=1$ Hint: $\color{white}{\text{Look carefully, the differences. Also is it a coincidence that there}}$ $ \color{white}{\text{a...