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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.

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 \( a\).

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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 \(a+b+c\)?

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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.
###### Image credit: Flickr Snap

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 \(\large x^2\).

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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 \(b\)?

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