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## Geometric Inequalities

Given 5 sticks of length 1, 3, 5, 9, and 10, how many distinct triangles can be formed? Learn the techniques and develop an intuition for working with geometric inequalities.

# Level 5

If $S=\left| \sqrt { { x }^{ 2 }+4x+5 } -\sqrt { { x }^{ 2 }+2x+5 } \right| \quad \forall \quad x\epsilon R\\$ Then find the maximum value of $${ S }^{ 8 }$$.

Details and Assumptions:

$$\left| a \right|$$ means modulus of $$a$$.

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 $$a+b+c$$?

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

In the middle of the wall opposite to Brilli's, rests a sugar crystal, one 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 $$\large x$$. Determine the value of $$\large x^2$$.

###### Image credit: Flickr Snap

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

$$\sqrt{\strut 49+a^2-7\sqrt2a}+\sqrt{\strut a^2+b^2-\sqrt2ab}+\sqrt{\strut50+b^2-10b}$$

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

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

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