Is Competition Problem on Constrained Inequality Relevant in Research? Many inequality problems on variable $x,y,z$ or $a, b$ come with constraints such as $ab = 1,xyz = 1, x+y+z = 1$. For example, Let $a$ and $b$ be positive real numbers with $a + b = 1$ then prove that - $$ \frac{a^2}{a + 1}+ \frac{b^2}{b + 1} ≥\frac1 3$$  (Hungary 1996).

Now I understand that contest-problem is good for exercise but what I am asking, is that are such inequality are useful in research? Or are such inequality found in research level mathematics?

This question is motivated by this post (click here).
Such problems are very useful in research, particularly when we want to optimize some property of a system under constraints.  This occurs when we're designing computer chips (optimize bandwidth consistent with power envelope), internal-combustion motors (optimize power consistent with fuel economy standards), finance (optimize return on investment consistent with portfolio or legal constraints), etc.
By C-S we obtain: $$\frac{a^2}{a+1}+\frac{b^2}{b+1}\geq\frac{(a+b)^2}{a+1+b+1}=\frac{1}{3}.$$
Now we see that $\frac{1}{3}$ is a minimal value of the function $f(a,b)=\frac{a^2}{a+1}+\frac{b^2}{b+1}$ 

under conditions $a&gt;0$, $b&gt;0$ and $a+b=1$ (because the equality occurs for $a=b=\frac{1}{2}$).

Id est, we solved an optimization problem without standard Lagrange Multipliers method. 

Specific this inequality is very easy, but of course  it can be found in research level mathematics.