Algebra
# Geometric Inequalities

$\large S=\left| \sqrt { { x }^{ 2 }+4x+5 } -\sqrt { { x }^{ 2 }+2x+5 } \right| \quad \forall x \in \mathbb R$

For $S$ as given above, find the maximum value of $S^8$.

**Notation:** $| \cdot |$ denotes the absolute value function.

Positive reals $x,y,z\ge \dfrac{\sqrt{3}}{3}$ satisfy the condition $xyz+x+y-z=0$. If $kxyz-xy-yz-zx\ge 1$ is always true, the the minimum value of $k$ can be expressed as $\dfrac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $a,c$ coprime and $b$ square-free.

What is $a+b+c$?

**12** feet high room is **40** feet long and **10** feet wide. Brilli the ant is standing in the middle of one of the walls (which is **10** by **12** feet), such that it is **1** foot above the floor, and is equidistant from the other **2** edges of that wall.

In the middle of the wall opposite to Brilli's, rests a sugar crystal, **1** foot below the ceiling. The minimum distance Brilli needs to cover to reach the sugar crystal, assuming that Brilli can walk anywhere inside the room, is $x$. Determine the value of $x^2$.

What is the smallest value (round off to 1 decimal point) that

$\sqrt{49+a^2-7\sqrt2a}+\sqrt{a^2+b^2-\sqrt2ab}+\sqrt{50+b^2-10b}$

can have for positive real numbers $a$ and $b$?