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}+\...