AoPS Academy
of positive integers that satisfy the equation![\[
x + xy + xyz = 31.
\]](http://latex.artofproblemsolving.com/a/7/8/a78d1e9d66831f11e7de08c109855e7727ec8535.png)
be real numbers such that 
, and![\[
a + \frac{1}{a} + b + \frac{1}{b} + c + \frac{1}{c} + d + \frac{1}{d} = 0.
\]](http://latex.artofproblemsolving.com/a/9/0/a90c94d82e3650560635616dacaf392d300b8c91.png)
Prove that one of the numbers 
or 
is equal to 
.
be an integer. Show that, if 
is an integer, then it is a perfect square.[/quote]
. If is an integer, then 
is at least rational, so that 
must be a perfect square then. Using Conway's topograph method, I have found out that the smallest non-negative pairs 
for which this happens are 
and 
, and that, for every such pair 
, the "next" such pair can be calculated as
The eigenvalues of that matrix are irrational, however, so that any calculation which uses powers of that matrix is a little cumbersome. There must be an easier way, but I cannot find it. Can you?
numbers placed in a circular way is either 
or 
. (Both values appear). Consider the total sum of the products of every 
consecutive numbers.
Find the minimal possible value of this sum.
Find the maximal possible value of this sum.
is a triangle with circumcenter 
and 
is a circle tangent to 
and internally to 
The tangent
to intersects again at 
If 
are the contact points of with 
respectively, prove that 
The correct is above. I fixed it.
be intersection of 2 circles and 
be intersection of 
and 
.
bisects 
.
and by ptolemy
.
be intersection of 
an
and 
are similar. By tangency on P, we get 
.