 # Please solve and explain the answer to two questions: Anonymous
timer Asked: Mar 24th, 2017
account_balance_wallet \$30

### 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. first_question.gif second_question.gif

### Unformatted Attachment Preview

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

�...

flag Report DMCA  Review Anonymous
awesome work thanks Brown University

1271 Tutors California Institute of Technology

2131 Tutors Carnegie Mellon University

982 Tutors Columbia University

1256 Tutors Dartmouth University

2113 Tutors Emory University

2279 Tutors Harvard University

599 Tutors Massachusetts Institute of Technology

2319 Tutors New York University

1645 Tutors Notre Dam University

1911 Tutors Oklahoma University

2122 Tutors Pennsylvania State University

932 Tutors Princeton University

1211 Tutors Stanford University

983 Tutors University of California

1282 Tutors Oxford University

123 Tutors Yale University

2325 Tutors