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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.

# Geometric Inequalities: Level 3 Challenges

For all rectangles with perimeter $$P$$ and area $$A$$, what is the minimum value of $$\dfrac{P^2}{A}$$?

If $$AB = 4$$, $$AC = 5$$, $$BD = 8$$, $$CD = 15$$, and $$AD$$ is a positive integer, find $$AD$$.

Over all real numbers $$x$$, find the minimum value of $$\sqrt{(x+6)^2+25} + \sqrt{(x-6)^2+121}$$.

Brilli the ant is trapped on a cube-shaped planet. She wants to go from one corner [point$$A$$] to the opposite corner [point $$B$$]. However she wants do this in a way such that she has to cross the shortest distance possible. If the length of the sides of the cube is $$1$$, the shortest distance between $$A$$ and $$B$$ can be expressed as $$p+\sqrt{q}$$ where $$p$$ and $$q$$ are non-negative integers and $$q$$ is square-free. What is $$q-p$$?

Details and assumptions:

Brilli the ant is completely confined to the surface of the cube. She can't move inside or outside the cube.

A triangle with integral sides has two sides equal to 1 unit. What is the measure of the angle included between the two sides?