### Question Description

Please take a look at the two GIFs that I have uploaded. I have a tutee who is stumped on two practice math questions, but I am unsure of how to explain them to her... or how to answer them.

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## Final Answer

here is my solution to 1st question

Question 3:

First, we will show that there exists natural number n for which no circle with lattice point center has

exactly n lattice points inside it.

If there is a circle with a lattice point as center and radius r ï‚£ 1 , then there is only 1 lattice point in its

interior (the center of the circle), however if the radius satisfies 1 ï€¼ r ï‚£ 2 then there lie exactly 5 lattice

points inside the circle. There is no circle with a lattice point as center where there would lie exactly 2, 3,

or 4 lattice points.

If there were a circle with radius r ï‚£

1

and center at the midpoint of a side of any square, then there

2

would be no lattice point in the circle. However, for a radius r satisfying

1

5

ï€¼rï‚£

, there would be

2

2

exactly 2 lattice points lying inside the circle.

If there were a circle with center at the center of any square and radius r ï‚£

no lattice points lying inside the circle. However, for r satisfying

2

, then there would be

2

2

10

ï€¼rï‚£

, then there would be

2

2

exactly 4 lattice points lying inside the circle.

If the center of the circle were slightly removed from the center of the square along a diagonal, then

taking the radius of the circle to be the distance of the new center from the farthest vertex of the square,

we would obtain a circle containing exactly 3 lattice points in its interior.

Now, we will prove that the plane can be turned about the circleâ€™s center so that with a suitable radius

r , there lies in the circleâ€™s interior any finite number of lattice points. We show that if the circleâ€™s center

ïƒ¦

ïƒ¨

1ïƒ¶

3ïƒ¸

is the point P with coordinates ïƒ§ 2, ïƒ· , then for every nïƒŽ

, there exists a radius rn such that inside

the circle centered at P and radius rn there lie exactly n lattice points.

Suppose that P1 and P2 are lattice points at the same distance from P . Let

ï€¨ a, b ï€© be the coordinates of

P1 and ï€¨ c, d ï€© those of P2 , where a, b, c, d are all integers. Since P1 and P2 are equidistant from P , the

square of their distances from P are equal. Therefore, by Pythagoreanâ€™s theorem, we have

ï€¨

aï€ 2

ï€©

2

2

ï€¨

1ïƒ¶

ïƒ¦

ï€...

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