[/quote]
,
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
is divisible with sum of all 
be a triangle with 
,
be the foot of the altitude from 
.
be a point in the interior of the segment 
.
be the point on the segment 
such that 
.
be the point on the segment 
such that 
.
be the point of intersection of 
and 
.
.
.
.
is selected in the line segment 
.
and 
intersect 
,
and 
intersect (a second time) the circumcircles to triangles 
,
passes through a fixed point 
distinct from 
.
,
,
,
for all 
,
be two positive sequences defined by 
and![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](http://latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
with 
,
and 
for all positive integers 
where 
is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)
Y by
Let 
and 
be positive integers. Let 
be the set of integer points 
with 
and 
. A configuration of 
rectangles is called 
if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
and 
be positive integers. Let 
be the set of integer points 
with 
and 
. A configuration of 
rectangles is called 
if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
case is trivial. Now, we induct proving that 
is odd if 
is odd.
t![[asy]
unitsize(0.5 cm);
dot((0,0), purple);
dot((1,0), black);
dot((2,0), black);
dot((3,0), purple);
dot((4,0), black);
dot((5,0), black);
dot((0,1), red);
dot((1,1), black);
dot((2,1), black);
dot((3,1), red);
dot((4,1), black);
dot((5,1), black);
dot((0,2), orange);
dot((1,2), black);
dot((2,2), black);
dot((3,2), orange);
dot((4,2), black);
dot((5,2), black);
dot((0,3), red);
dot((1,3), black);
dot((2,3), black);
dot((3,3), red);
dot((4,3), black);
dot((5,3), black);
dot((0,4), orange);
dot((1,4), black);
dot((2,4), black);
dot((3,4), orange);
dot((4,4), black);
dot((5,4), black);
dot((0,5), purple);
dot((1,5), black);
dot((2,5), black);
dot((3,5), purple);
dot((4,5), black);
dot((5,5),black);
[/asy]](http://latex.artofproblemsolving.com/b/1/4/b147c3592d18385013298e97720574f5e33fe125.png)
ways to pair up the 2n points. From here, there are 
to partition the remaining points into rectangles. Both of these quanitities are odd so total ways in this case are odd.![[asy]
unitsize(0.5 cm);
dot((0,0), purple);
dot((1,0), black);
dot((2,0), purple);
dot((3,0), black);
dot((4,0), black);
dot((5,0), black);
dot((0,1), red);
dot((1,1), black);
dot((2,1), black);
dot((3,1), black);
dot((4,1), red);
dot((5,1), black);
dot((0,2), orange);
dot((1,2), black);
dot((2,2), black);
dot((3,2), orange);
dot((4,2), black);
dot((5,2), black);
dot((0,3), red);
dot((1,3), black);
dot((2,3), black);
dot((3,3), black);
dot((4,3), red);
dot((5,3), black);
dot((0,4), orange);
dot((1,4), black);
dot((2,4), black);
dot((3,4), orange);
dot((4,4), black);
dot((5,4), black);
dot((0,5), purple);
dot((1,5), black);
dot((2,5), purple);
dot((3,5), black);
dot((4,5), black);
dot((5,5), black);
[/asy]](http://latex.artofproblemsolving.com/5/9/6/5965fa66d20c09cf6a622aebf1bebe9392421668.png)
![[asy]
unitsize(0.5 cm);
dot((0,0), purple);
dot((1,0), black);
dot((2,0), black);
dot((3,0), black);
dot((4,0), purple);
dot((5,0), black);
dot((0,1), red);
dot((1,1), black);
dot((2,1), red);
dot((3,1), black);
dot((4,1), black);
dot((5,1), black);
dot((0,2), orange);
dot((1,2), black);
dot((2,2), black);
dot((3,2), orange);
dot((4,2), black);
dot((5,2), black);
dot((0,3), red);
dot((1,3), black);
dot((2,3), red);
dot((3,3), black);
dot((4,3), black);
dot((5,3), black);
dot((0,4), orange);
dot((1,4), black);
dot((2,4), black);
dot((3,4), orange);
dot((4,4), black);
dot((5,4), black);
dot((0,5), purple);
dot((1,5), black);
dot((2,5), black);
dot((3,5), black);
dot((4,5), purple);
dot((5,5), black);
[/asy]](http://latex.artofproblemsolving.com/0/1/9/019d37253bfb07489ae46035b82a752909428d06.png)
)
not-necessarily-distinct configurations. The only class of configurations with odd size is the one with 
