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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?

Give your answer in degrees.


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