Please solve and explain the answer to two questions:

Anonymous
timer Asked: Mar 24th, 2017
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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.

Please solve and explain the answer to two questions:
first_question.gif
Please solve and explain the answer to two questions:
second_question.gif

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Tutor Answer

Ace_Tutor
School: UT Austin

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  1 , then there is only 1 lattice point in its
interior (the center of the circle), however if the radius satisfies 1  r  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 

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

2
, then there would be
2

2
10
r
, 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 circle’s center so that with a suitable radius
r , there lies in the circle’s interior any finite number of lattice points. We show that if the circle’s center




1
3

is the point P with coordinates  2,  , then for every n

, 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

 a, b  be the coordinates of

P1 and  c, d  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 Pythagorean’s theorem, we have



a 2



2

2



1

...

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Anonymous
awesome work thanks

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