Quiz 2: Domain and graph of |x-y²|=1-|x|

Quiz 1: Find The Minimum Value Of An Expression

$|x+|x|+m|+|x-|x|-m|=2$

Well, on my last blog post, I have shown with a strong Olympiad Math competition problem about how useful we have to always bear in mind that when $x$ is an solution to the function, says, $P(x)$, then we need to consider if $-x$ could also be a solution the that function of $P(x)$.

Fortunately that is not something that is hard to check. All that we need to do is to determine if $P(x)=P(-x)$.

I will show with one more challenging, hard and intriguing math competition problem why remembering to check with this fact could save our day from struggling with the problem.

Find all numbers of $m$ such that the equation $|x+|x|+m|+|x-|x|-m|=2$ has exactly three solutions.

Find The Unknown Function...

Let $f(x)$ be an unknown function defined on $[0,\infty)$ with $f(0)=0$ and $f(x)\le x^2$ for all $x$. For each $0\le t$, let $A_t$ be the area of the region bounded by $y=x^2$, $y=ax^2$ (where $1<a$) and $y=t^2$. Let $B_t$ be the area of the region bounded by $y=x^2$, $y=f(x)$ and $x=t$. See the image below:

























a) Show that if $A_t=B_t$ for some time $t$, then:

[MATH]\int_0^{t^2}\sqrt{y}-\sqrt{\frac{y}{a}}\,dy=\int_0^t x^2-f(x)\,dx[/MATH]

b) Suppose $A_t=B_t$ for all $0\le t$. Find $f(x)$.

c) What is the largest value that $a$ can have so that $0\le f(x)$ for all $x$?