Geometric Inequalities

Geometric Inequalities: Level 5 Challenges


S=x2+4x+5x2+2x+5xR \large S=\left| \sqrt { { x }^{ 2 }+4x+5 } -\sqrt { { x }^{ 2 }+2x+5 } \right| \quad \forall x \in \mathbb R

For SS as given above, find the maximum value of S8S^8.

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

Positive reals x,y,z33x,y,z\ge \dfrac{\sqrt{3}}{3} satisfy the condition xyz+x+yz=0xyz+x+y-z=0. If kxyzxyyzzx1kxyz-xy-yz-zx\ge 1 is always true, the the minimum value of kk can be expressed as abc\dfrac{a\sqrt{b}}{c} for positive integers a,b,ca,b,c with a,ca,c coprime and bb square-free.

What is a+b+ca+b+c?

A 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 xx. Determine the value of x2 x^2.

Image credit: Flickr Snap

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


can have for positive real numbers aa and bb?

Find the square of the minimum value of 9+x2+(4x)2+(3y)2+4+y2\sqrt{9+x^{2}}+\sqrt{(4-x)^{2}+(3-y)^{2}}+\sqrt{4+y^{2}} as xx and yy range over all real numbers.


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