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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