Day two of that Putnam
by Wolstenholme, Aug 8, 2015, 7:35 AM
Let 
, 
, 
be integers for 
. Assume for each 
, at least one of , , is odd. Show that there exists integers 
such that 
is odd for at least 
values of , .
except for 
and count the number of times 
odd results in total and so by the pigeonhole principle one of your seven choices of 
works. Probabilistic method yay!
Prove that the expression ![\[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \]](http://latex.artofproblemsolving.com/c/0/a/c0a6877cd49e904728b24e7743fe8021efd6d312.png)
is an integer for all pairs of integers 
.
such that 
so our expression is equal to![\[ \frac{am + bn}{n}\binom{n}{m} = b\binom{n}{m} + a\binom{n - 1}{m - 1} \]](http://latex.artofproblemsolving.com/0/6/d/06d84438ad75704bc64860571133424d4385b0f5.png)
Let 
be a continuous function such that 
for all 
. Show that 
for 
.
we have that 
and so since 
is continuous we must have 
.
.
be a number such that 
.
and iterating we have that if 
for any 
then 
.
for 
are clearly dense in 
(since we can make 
arbitrarily large) so since 
by continuity we have that ![$[-1, 1]$](http://latex.artofproblemsolving.com/5/c/3/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png)
as desired.
Let 
be a finite set of positive integers. We define finite sets 
of positive integers as follows: the integer 
in 
if and only if exactly one of 
or is in 
. Show that there exist infinitely many integers 
for which 
.
and let 
.
hence 
.
which is clearly equivalent to the problem condition so since there are infinitely many powers of two we are done.
Let 
be a set of more than 
distinct points with coordinates of the form 
in 
-dimensional space with 
. Show that there are three distinct points in which are the vertices of an equilateral triangle.
consider the 
away from each other. Now we note that the expected value of 
so for some 
.Well I'd say I got at least a 101 on this Putnam!!!! The overall first place score in 2000 was 96....darn wish I could take it haha
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