# Squaring a circle, again

Geometry Level 5

There is a unit circle with its center point at $$(0,0)$$ . There is a set of 4 linear functions in the form: $\dfrac{m}{n} x , -\dfrac{m}{n} x , \dfrac{n}{m} x , -\dfrac{n}{m} x$ which intersect the circle at 8 points. There is a square that intersects the unit circle at the same 8 points with sides parallel to X and Y axis.
If $$s$$ is semi-perimeter of the square then $$\lfloor { s \times 10^9 } \rfloor = \lfloor {\pi \times 10^9 } \rfloor$$.

Find the set of smallest positive integer numbers $$m < n$$ such that they meet the described criteria. Find all the prime factors of numbers $$m$$ and $$n$$. For each factor $$p_{i}$$, find $$s_{i}$$ which is number of primes equal to or less then $$p_{i}$$. Give answer as a product of all $$s_{i}$$.

Details and Assumptions:

If some prime number appears more than once as a factor use it only once in the solution, i.e. if $$m$$ has prime factors: $$2^2, 3, 5$$ and $$n$$ has prime factors: $$2^3, 5^2, 7$$, use only numbers $$2, 3, 5, 7$$ in calculating the final answer which in this case would be $$1\times 2\times 3 \times 4 = 24$$. The problem is original.

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