## 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 \(A\)? | |
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Part B | 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 \(N=8\) | |
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Part B | Determine \(D\) when \(N=12\) |