Geometric Inequalities

Geometric Inequalities: Level 3 Challenges


For all rectangles with perimeter PP and area AA, what is the minimum value of P2A?\frac{P^2}{A}?

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

Over all real numbers xx, find the minimum value of (x+6)2+25+(x6)2+121 \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 [pointAA] to the opposite corner [point BB]. 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 11, the shortest distance between AA and BB can be expressed as p+qp+\sqrt{q} where pp and qq are non-negative integers and qq is square-free. What is qpq-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?

Give your answer in degrees.


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