# Computation Workshop Solution Checker

## Lattice Points in Circles

A Lattice Point is any point $$(x,y)$$ in the Cartesian Plane such that both $$x$$ and $$y$$ are integers (note that either $$x$$ or $$y$$ or both may be negative). Let $$R$$ and $$\lambda$$ be two fixed numbers. Dr Cartesian has drawn $$2$$ circles, both of radius $$R$$, in the Cartesian Plane, centred at points $$(-\lambda,0)$$ and $$(\lambda,\lambda)$$ respectively. Let $$P$$ be the number of lattice points that lie inside exactly one of the two drawn circles (lattice points that are precisely on the circumference of a circle do not count as being inside the circle). Dr Cartesian wants to know the value of $$P$$ for various values of $$R$$ and $$\lambda$$. For example, when $$\lambda=1$$ and $$R=2$$, Dr Cartesian calculates that $$P=14$$.

Part A When $$\lambda=123$$ and $$R=234$$, what is $$P$$? When $$\lambda=123456$$ and $$R=234567$$, what is $$P$$?

## Linexponential Series

For each positive integer $$N$$ let

$$S(N) = 1 + 2 \cdot 3 + 3 \cdot 3^2 + 4 \cdot 3^3 + 5 \cdot 3^4 + \cdots + (N+1) \cdot 3^N$$

So for example $$S(1) = 1 + 6 = 7$$ and $$S(2) = 1 + 6 + 27 = 34$$.

Part A What are the last six digits of $$S(123)$$? What are the first six digits of $$S(123456789)$$? What are the last six digits of $$S(2021^{2021})$$?