Approach to inequality proofs

contest-mathinequalityproblem solving

I have read much of the theory about inequalities but cannot solve sums from them of Maths Olympiad. What should be my general approach to solve them. I can share a few sums.

Let a, b, c be positive real numbers such that
$\frac
{a}{1+a}+\frac{b}{1+b}+\frac
{c}{1+c}=1$.
Prove that $abc ≤ 1/8$.

$a,b,c$ are real number with $a+b+c=1$
Prove that $\sum_{cyc}{\frac{a}{a²+b³+c³}}≤\frac{1}{5abc}$

I do not need solution to these sums as I have the solutions. But I am unable to solve inequality sums of this kind. What should be my strategy to solve these sums? Any suggestion will be extremely helpful.

Best Answer

I think, I understand your problem: you say that you fully understand solutions, but you can't solve a new problem by yourself. It happens because you don't try to solve a problem with a "fully understandable solution".

Just try to solve a problem, for which a solution you think that you understood, and write this solution!

For example. Take a problem from the Dietrich Burde's link:

Let $x,$ $y$ and $z$ be positive numbers, such that $x+y+z=3$ . Prove that :

$$\sum_{\mathrm{cyc}}\frac{x}{x^3+y^2+z} \leq 1.$$

  1. Try to solve this problem by yourself without to see a solution, although you are convinced that you have seen the proof and understood it.

  2. If you can't find a solution, so read a solution and go back to 1.

  3. If you think that you know already a solution of the problem, so write a full solution.

  4. If you can't write a full solution, so go back to 1.

Try to repeat these things until the moment, that you succeed to write a full solution.

This is a first step. After this you can try to understand, how we can solve this problem by another way, how we can guess the solution, how we can create a similar problem and more.

Do it with all problems and you'll start to solve problems by yourself. Good luck!

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