The ONLY problem I lost points on - BIH JBMO TST 2013 - Problem 2

LaTeX:
Let $a$, $b$ and $c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove the following inequality:
$\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}$

[محمدطلعت-ذع]

Great job, your videos are most satisfying + at 3:07 we can also use the QM-AM inequality and the fact that a^2 + b^2 + c^2 = 3 to be finish the problem more quickly.

[alesb]

I have a question: do you think developing problem-solving skills could be useful, despite attending the last year of high school and having poor experience (mainly due to the past lack of attention or interest) with Olympiads?

[Szynkaa]

i did it slightly different way- using Cauchy-Schwartz inequality with a_1=a, a_2=b, a_3=c and b_1=(1/(3c(a^2-ab+b^2)), b_2 b_3 similiarly. Then we apply our condition a^2+b^2+c^2=3 and inequality we have to prove is already homogenous. Then some simple calculations and we get cyclic sum of (a^2*b^2/(a^2-ab+b^2)^2) <=3 which is truth as its easy to see every part of the sum is <=1.

[quite_unknown_]

For the last part you could Also instantly use AM-QM

[AnuragSingh-monb]

i can relate for losing points on an inequality, so here's sympathy from my part

[rtl]

I have a question: are AMC 2022-2023 exams going to be held online? And also could you please tell me a way to prepare for math olympiad as a beginner? Actually this is my last chance and also my first math olympiad..i live in bangladesh..so to do well in selection stage what should i do? I have been preparing from april and learnt lots of new things but still i feel like my problem solving ability isn't that strong..could you please help me with some suggestions? Thank you in advance