2018-05-07 Advanced

         

The centers of three identical coins form the angle that's colored green above.

What angle maximizes the area of the blue convex hull?

Note: You can imagine the perimeter of the convex hull as a rubber band stretched around the three coins.

How many triangles can you make using points on this \(5\times 5\) square grid as their vertices?

Note: Some of the triangles can be congruent, and they can overlap one another.

Three possible triangles are shown.

Three possible triangles are shown.

\[\large \sum_{n=1}^{\infty}\frac1{n2^n}=\, ?\]

A wire in the shape of the curve \(y = x^2\) carries a current \(I.\) It also contains some horizontal, semi-infinite segments.

The magnitude of the magnetic flux density \((B)\) at the point \((x,y) = (0,1)\) is \[B = \alpha \, \frac{\mu_0 \, I}{4 \pi}.\] If \(\alpha\) is a positive real number, what is its value?

The figure shows part of an infinitely large cube lattice.

What percentage of points are visible from a given point in the infinite lattice (to the nearest integer)?

Note: A point is not visible if it is directly behind another point in the line of sight.

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