Sunaina thinks absurd

Okie!! Fine, I am late by 2 weeks but I was busy in OTIS submissions. And yayy!! I learnt how to use Evan.sty .  These problems are the exercises  from a Titu handout in  this website .  Here's the full  magazine , go to page 40's and one can find it :) So full Titu :P 10th position ( IMO shortlist, 1996) :  Suppose that $a, b, c > 0$ such that $abc = 1$. Prove that $$\frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1. $$ Walkthrough:  Thanku  Rohan Bhaiya 😄   a.  $$\sum_{cyc} \frac{ab}{a^5+b^5+ab}\le \sum_{cyc}\frac{c}{a+b+c}=1.$$ b. Cross Multiplying, it is enough to show that $$a^2b+ab^2+abc\le a^5c+b^5c+abc . $$ c. Multiply $abc=1$ to each side and use Muirhead. 9th position (RMO, 2006): If $ a,b,c$ are three positive real numbers, prove that $$ \frac {a^{2}+1}{b+c}+\frac {b^{2}+1}{c+a}+\frac {c^{2}+1}{a+b}\ge 3$$ Walkthrough: a. Using Titu, get $$\frac {a^{2}+1}{b+c}+\...

 This week was full algebra biased!! 😄  This is my first time trying inequalities, so this pure beginners level.   Do try all the problems first!! And if you guys get any nice solutions , do post in the comments section!  These problem uses only  Power mean Inequality   and  Titu's lemma . The First few problems happen to be  not problems, but tricks(?) which are extensively used..  Here are the walkthroughs of this week's top 10 Inequality problems! 10th position:  Prove that for any real $a>0$ , $a+\frac{1}{a}\ge 2$ Walkthrough: a.  only AM-GM 9th position: Prove that for any real  $a>0$ , $\frac{a}{1+a^2}\le \frac {a}{2a}$. Walkthrough: a. Only AM-GM  b. Use AM-GM to show that   $1+a^2\ge 2a $  8th position:  Prove that for any real $x,y>0$ ,$\frac{1}{x+y}\le \frac{1}{4x}+\frac{1}{4y}$ Walkthrough: a. AM-HM inequality (cute 💖)  7th position :  Prove that for any real positiv...