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Regular Season Problem 11 from 2023 MIT Integration Bee

Prove $\sum_{cyc}xy\left(2\sqrt{z^2+1}-z\right)\ge\sum_{cyc}\left(\sqrt{z^2+1}-z\right)$ when $xy+yz+zx=1$

How to prove $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{16(ab+bc+ca)^2}\ge \frac{1}{a+b+c}.\left(\dfrac{1}{\sqrt{abc}}-3\right)$

If $a^2+b^2+c^2+abc=4$, Find minimum $P=\sqrt{\frac{2a+bc}{3}}+\sqrt{\frac{2b+ca}{3}}+\sqrt{\frac{2c+ab}{3}}-\frac{3(a+b+c+abc)}{2}$

Alternate method to solve this integral

Integrate $\frac{x}{\sqrt{1-x^2}}$

using induction to prove $2^n \geq n + 5$

Regarding improper integral and normal distribution

Properties of substitution system

Why does sometimes the value of definite integral becomes $0$ for a non zero graph of a function

Completing the square under a square root

Verification of solution by $u$-substitution of $\int \frac{\ln x}{1+x^2}dx$

$u$-substitute in integration

Solving a polynomial equation using substitution

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