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Let the touchpoint of the incricle with $BC$ be $F$. As $DP \perp IM$, $IP^2-PM^2=ID^2-DM^2$ And as $EP \perp IN$, $PN^2-PI^2=EN^2-EI^2$ Summing these two, we get $PN^2-PM^2=(EN^2-DM^2)+(ID^2-IE^2)$ $PN^2-PM^2=(EN^2-DM^2)+(IC^2-IB^2)$ [Note that $ID=IC$,$IE=IC$] $PN^2-PM^2=(EN^2-DM^2)+(CF^2-BF^2)$ [...

This problem is just a mere application of only one lemma.The perpendicular lemma.Use the lemma two times only.

না ভাই।করি নাই।করার চেষ্টা করতেছি।করলে দিব সল্যুশান।

Let $n \geq 1$ be an odd integer.Determine all functions $f$ from the set of integers to itself such that for all distinct integers $x$ and $y$,$f(x)-f(y)|x^{n}-y^{n}$.

Find all positive integers $m,n \geq 2$,such that,
$1$. $m+1$ is a prime number of the form $4k-1$
$2$.there is a prime number $p$ and non-negative integer $a$,such that
\[\frac{m^{2^{n}-1}-1}{m-1}=m^{n}+p^{a}\]

I wish people started posting cool medium level number theory problems like this.

Let $n$ and $p$ be integers such that $n>1$ and $p$ be a prime.If $n|p-1$ and $p|n^3-1$,show that $4p-3$ is a perfect square.

My solution is same.

Well, I meant $a,b \in N_0$. :mrgreen: Sorry for the typo. :mrgreen: I was searching for a title.But i couldn't find anything fitting with this problem.And i don't know the source either.So i gave this title on a whim.Because my forum favourite emo is mrgreen. :mrgreen: :mrgreen: :mrgreen: :mrgreen:...