# Graph, then count (?)

Geometry Level 4

$$1.$$ Find the number of integral points inside a circle (excluding the boundary) of a circle with equation $\displaystyle x^2 + y^2 = 225.$

$$2.$$ If the answer in the question above is $$N$$, find the remainder when $$N^{1000}$$ is divided by $$9.$$ Then,

$$3.$$ If the answer in $$2$$ is $$M$$, find the value of the infinite sum $1 + 2\left(\frac{1}{M} \right) + 3 \left(\frac{1}{M^2}\right) + 4\left(\frac{1}{M^3}\right) + ...$

Details and assumptions

To get the correct answer, first answer $$1$$, and with that value you've got, use that to answer $$2$$. Finally, from the answer in $$2$$, use that to answer question $$3.$$ Round your final answer to the nearest integer.

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