Well I did a Putnam for fun... and I did unexpectedly well
by Wolstenholme, Aug 7, 2015, 7:47 AM
So, here's what I could accomplish today on the 2000 Putnam
Let 
be a positive real number. What are the possible values of 
given that 
are positive numbers for which 
?
Prove that there exist infinitely many integers 
such that , 
, 
are each the sum of the squares of two integers. [Example: 
, 
, 
.]
The octagon 
is inscribed in a circle with the vertices around the circumference in the given order. Given that the polygon 
is a square of area 
, and the polygon 
is a rectangle of area 
, find the maximum possible area of the octagon.
Show that the improper integral ![\[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \]](//latex.artofproblemsolving.com/8/3/5/83584525a5af2b8544d1c2fa084b81f41ebdf47d.png)
converges.
Three distinct points with integer coordinates lie in the plane on a circle of radius 
. Show that two of these points are separated by a distance of at least 
.
Let 
be a polynomial with integer coefficients. Define a sequence 
of integers such that 
and 
for all 
. Prove that if there exists a positive integer 
for which 
then either 
or 
.
Let 
be a positive real number. What are the possible values of 
given that 
are positive numbers for which 
?![\[ 0 < \displaystyle\sum_{j=0}^{\infty} x_j^2 < \left(\displaystyle\sum_{j=0}^{\infty} x_j\right)^2 = A^2. \]](http://latex.artofproblemsolving.com/8/e/3/8e3729694c0748f6ae69708c19dd403aa7284bb6.png)
let 
for some 
.![\[ \displaystyle\sum_{j=0}^{\infty} x_j^2 = A^2\left(\frac{1 - r}{1 + r}\right)\]](http://latex.artofproblemsolving.com/6/5/1/651310b387cdccdff605ffcea21922b3d3761882.png)
takes on all values in the interval 
we have that the desired interval is precisely 
Prove that there exist infinitely many integers 
such that , 
, 
are each the sum of the squares of two integers. [Example: 
, 
, 
.]
for some integer 
are clearly the sum of two squares, so it suffices to show that there exist infinitely many 
is the sum of two squares. I claim even more - there exist infinitely many 
for some positive integer 
.
and note that 
is a solution. Hence with standard Pell equation theory every pair 
where 
for positive integers 
is a solution and hence we have found infinitely many 
The octagon 
is inscribed in a circle with the vertices around the circumference in the given order. Given that the polygon 
is a square of area 
, and the polygon 
is a rectangle of area 
, find the maximum possible area of the octagon.
so if 
are the length and width of the rectangle then 
and 
.
)
and 
.
and 
and 
and 
.
with 
.
and 
and 
.![\[ [P_2P_3P_4] = [P_6P_7P_8] = \frac{\sqrt{2}}{2}\left(x - \sqrt{2}\right) \]](http://latex.artofproblemsolving.com/2/7/e/27ed8cfe15c413694755e20e00e592ded41c450f.png)
![\[ [P_4P_5P_6] = [P_8P_1P_2] = \sqrt{2}\left(x - \frac{\sqrt{2}}{2}\right) \]](http://latex.artofproblemsolving.com/2/e/4/2e4ae28f95f98415d5edc3771352dfdae52104c3.png)
![\[ 3\sqrt{2}x. \]](http://latex.artofproblemsolving.com/2/a/5/2a544355dbac8c7ba1d1faf6bc603bdf5c7d2e22.png)
(the radius of the circle) hence the maximum area of the octagon is 
.
Show that the improper integral ![\[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \]](http://latex.artofproblemsolving.com/8/3/5/83584525a5af2b8544d1c2fa084b81f41ebdf47d.png)
converges.
and 
and that 
converges.
Three distinct points with integer coordinates lie in the plane on a circle of radius 
. Show that two of these points are separated by a distance of at least 
.
and that the triangle doesn't contain the center of the circle (or else one side would subtend an arc greater or equal to one-third the circle's circumference and we would be trivially done). Let 
be the longest side of our triangle. It's clear then that the altitude to side 
so the area of the triangle if equal to or less than![\[ \frac{s}{2}\left(r - \sqrt{r^2 - \frac{s^2}{4}}\right) = \frac{s^3}{8\left(r + \sqrt{r^2 - \frac{s^2}{4}}\right)} \le \frac{s^3}{8r} \]](http://latex.artofproblemsolving.com/b/1/6/b16bd4ab9c9656f637b1a94703db6e9bd99cf5e7.png)
hence we have 
and thus ![$s \ge \sqrt[3]{4r} > \sqrt[3]{r}$](http://latex.artofproblemsolving.com/f/3/b/f3bff6c7d1601eb8788debdf92be2bfeeb160370.png)
as desired.
Let 
be a polynomial with integer coefficients. Define a sequence 
of integers such that 
and 
for all 
. Prove that if there exists a positive integer 
for which 
then either 
or 
.
so since 
and since the sequence of differences 
is also purely periodic each difference must be equal in absolute value to 
.
holds. Then it's clear by periodicity that 
must all be distinct and now it's easy to see that we must have 
as desired or else the previous statement will be contradicted.This post has been edited 2 times. Last edited by Wolstenholme, Aug 8, 2015, 4:18 AM
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