One definite integral problem

Let $a,b,c\in\mathbb{Z}$, $1<a<10$, $c$ is a prime number and $f(x)=ax^2+bx+c$. If $f(f(1))=f(f(2))=f(f(3))$, find $f'(f(1))+f(f'(2))+f'(f(3))$

Combinatorial proof of ${2n\choose n}n!\le (2n)^{n}$

For what $a$ and $b$ are there explicit expressions for $I(a, b) =\int_0^1 \int_0^1 \dfrac{dx\,dy}{1-x^ay^b} $?

Chi Squared Clarification

$\int x^{2}\sqrt{a^{2}+x^{2}}\,dx$. Is there another way to solve it faster?

Find a minimal sufficient statistic for $U(\theta,\theta+c)$, where $(\theta,c)$ unknown.

Given a finite number of stones. We place every stone on an integer. Prove that, given different movements, we can only make finite number of moves

Is there a direct proof for $\int_0^{2\pi}\frac{r^2+r(\cos t-\sin t)}{1+2r\cos t+r^2}dt=2\pi$

Proof of $BAC-CAB$ identity missing step

Cat and Mice puzzle!

Prove the roots of $z^7+7z^4+4z+1=0$ lie inside a circle of radius $2$.

Infinite Rational numbers add up to give irrational numbers.

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