# Computation Workshop Solution Checker

## Smallest Real Number

Let $$A$$ be the smallest real number such that $$\frac{p^2+2q^2}{pq} \geq A$$ for all positive integers $$p$$ and $$q$$.

Let $$B$$ be the smallest real number such that $$\frac{p^5+2q^5+3r^5+4s^5+5t^5}{pqrst} \geq B$$ for all positive integers $$p$$, $$q$$, $$r$$, $$s$$, and $$t$$.

Part A What are the $$6$$ digits immediately to the right of the decimal point of $$A$$? What are the $$6$$ digits immediately to the right of the decimal point of $$B$$?

## Diagonal Polygons

Let $$N$$ be a fixed positive integer. A lattice point is any point $$(x,y)$$ in the Cartesean plane such that both $$x$$ and $$y$$ are integers with $$0 \leq x < N$$ and $$0 \leq y < N$$. A polygon is called a `Diagonal Polygon' if all of its vertices are lattice points and all its side-lengths are exactly $$\sqrt{2}$$. Let $$D$$ denote the total number of diagonal polygons. For example when $$N=5$$, we have $$D=10$$ (nine of them are squares and the last one is a Greek cross shape).

[Note that the boundary of a polygon is not allowed to self-intersect]

Part A Determine $$D$$ when $$N=8$$ Determine $$D$$ when $$N=12$$