[/quote]
,
be a positive integer and 
a prime divisor of 
,
.
or 
,
is integer.
of residues modulo 
if
there is a nonzero residue 
modulo 
(all considered mod 
(not necessarily distinct) such that 
.
,
.
,
is marked. Point 
is on segment 
,
is on segment 
such that the circumcircles of triangles 
and 
are tangent to line 
.
and 
passing through the center of the circumcircle of triangle 
,
.
be a positive integer. Show that the numbers![\[ \binom{2^n - 1}{0},\; \binom{2^n - 1}{1},\; \binom{2^n - 1}{2},\; \ldots,\; \binom{2^n - 1}{2^{n - 1} - 1}\]](http://latex.artofproblemsolving.com/2/a/3/2a3629f4a907a9be48ba5faa50dfbf067252d3d6.png)
to 
,
,
,
,
in some order.
be an acute triangle with incentre 
and 
.
and 
intersect the circumcircle of 
and 
,
and 
and 
are parallelograms (with 
,
)
be the point of intersection of lines 
and 
.
,
positive real numbers such that 
. Prove that :
and 
,
such that![\[
\sqrt{n + a} + \sqrt{n + b} \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/5/4/d/54d20546de5736770a9929ac86163e3b06553b7d.png)
![\[
x = \sqrt{n + a} + \sqrt{n + b} \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/6/6/8/668d53723d307556536736fda0e4a19be848834f.png)
![\[
x^2 = (\sqrt{n + a} + \sqrt{n + b})^2 = (n + a) + (n + b) + 2\sqrt{(n + a)(n + b)}.
\]](http://latex.artofproblemsolving.com/1/9/f/19f2627a432354a757d3c9e8ba42e96f83f46726.png)
So:![\[
x^2 = 2n + a + b + 2\sqrt{(n + a)(n + b)}.
\]](http://latex.artofproblemsolving.com/0/7/3/073602693604bbf3cb1f5c1c2e5051e8994a5e78.png)
![\[
\sqrt{(n + a)(n + b)} \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/1/1/4/11441fab3ba768fee7a0eef4dbf68124d6637aec.png)
![\[
(n + a)(n + b) = k^2.
\]](http://latex.artofproblemsolving.com/0/e/c/0ec5e7bac2aebe9ec507727ed21f7ca028655222.png)
Assume 
.![\[
n + a \mid k \quad \text{and} \quad k \mid n + b,
\]](http://latex.artofproblemsolving.com/7/5/f/75fc52b6662df3a4c0cb2f40f49c4b3351c90a4d.png)
or it could also be that 
.![\[
(n + a)k_1 = k \quad \text{and} \quad kk_2 = n + b.
\]](http://latex.artofproblemsolving.com/1/f/5/1f5936241f7bebcf15c68fed5267bbfad2d40c2b.png)
![\[
k_1 k_2 = \frac{n + b}{n + a}.
\]](http://latex.artofproblemsolving.com/d/4/c/d4c1958e3554b58c3def282070f849842cc10912.png)
,![\[
k_1 k_2 = 1 + \frac{b - a}{n + a}.
\]](http://latex.artofproblemsolving.com/3/0/6/306a6a00c44d4e4db1b36c7e588f684f28fdbf44.png)
,
must be an integer, which is not possible.
be a sequence of positive integers with 
,
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
.
for some positive integer 
such that 
.
and 
,
,
be the tangency points of 
,
,
and let 
be the reflection of 
intersects the tangents to 
and 
.
.
such that![\[f(x+y+f(y))=f(x)+f(ay)\]](http://latex.artofproblemsolving.com/7/5/d/75d1ac71fa0c2e53fdfc8e198b109d18e935fe74.png)
holds for all 
.
of the integers with the following property: for any integer 
there is exactly one solution of 
with 
.
Y by Adventure10, megarnie, GeoKing
Suppose 
is a connected graph with 
edges. Prove that it is possible to label the edges 
in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.
Note: Graph-Definition. A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices
belongs to at most one edge. The graph 
is connected if for each pair of distinct vertices 
there is some sequence of vertices 
such that each pair 
is joined by an edge of 
.
is a connected graph with 
edges. Prove that it is possible to label the edges 
in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.Note: Graph-Definition. A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices
belongs to at most one edge. The graph 
is connected if for each pair of distinct vertices 
there is some sequence of vertices 
such that each pair 
is joined by an edge of 
.This post has been edited 1 time. Last edited by orl, Aug 15, 2008, 1:51 PM
(mario pickavet, bronze '91)
be the vertices involved in the path. If 
,
be another vertex. We can find a path 
,
,
are not among the 
'
,
will be equal to 
Otherwise, label the edges 
Since 
we are done.
s.t. 
are connected with each other and 
.
as 
as 
and 
respectively and 
as 4. Look at the vertex 
,
!?
such that 
and path 
(WLOG 
)
,
in succession. Then, each vertex is labeled either with 2 consecutive integers or with 
where we add edges to 
between pairs of odd-degree vertices. Then, 
such that each walk starts and ends at odd-degree vertices and each odd-degree is the endpoint of exactly one walk.
,![$[1, k_1]$](http://latex.artofproblemsolving.com/d/1/1/d119cc6926863efd3a3bafc64a8a81c175be6c2c.png)
in order, the second one with ![$[k_1+1, k_1+k_2]$](http://latex.artofproblemsolving.com/c/e/b/ceb9dee824e705b391db60eddfc1d78a53b533d9.png)
,![$[k_1+\cdots+k_{m-1}+1, k_1+\cdots+k_m]$](http://latex.artofproblemsolving.com/3/7/c/37c0ed81a5f949ec1189960c87b28beef99e05bb.png)
.
of edge labels equal to one. 
and 
to those first and last edges and of course the may not be coprime.
be the vertices of odd degree of 
,
in order. Otherwise, we will add the edges 
Let this modified graph be 
Note that this may result in the graph not being simple anymore, but that's fine since the problem still holds. Now, observe that 
from the cycle would partition the cycle into walks from 
to 
,
to 
,
,
to 
This makes it such that none of the walks can be cycles since the 
'
?
,
.
largest integers of 
where 
has an edge not in 
.
and replacing the edge 
with 
.
(with base case 
)
,
,
.
.
for which 
,
,
,
for which 
,
,
.
with 
with 
,
with 
.
for 
differ by 
.
edges, and is obviously still connected, unless it is empty. If it is empty then 
,
,
,
so that the condition is satisfied. Now, consider these labelings in conjunction with our labelings of the other edges. We know that the labelings are fine for 
since they do not change based upon the labelings of the induced graph since they are only adjacent to other 
'
.
,
for which 
,
.
in 
in a graph). Let 
be a cycle in 
.
with 
,
be the smallest such index for which 
,
be the second-smallest such index for which 
.
.
,
by minimality of 
with 
with 
with 
,
(these vertices have degree 
edges. Thus, by the inductive hypothesis we can label the edges 
so that the condition is satisfied for the reduced graph. Now, consider the entire graph. Then, for each vertex 
,
and 
,
be the vertices in the cycle. Then, label 
with 
,
with 
.
,
to get a connected graph on 
edges, so we can label the edges 
so that it works on the reduced graph. Now, it is not hard to see that it works in the original graph -- for all vertices which are not a 
,
(so it is 
)
.
for which 
,
.
with 
with 
,
with 
.
,
,
,
edges, so by the inductive hypothesis we can label it and then this labeling works for 
'
and 
all work since their neighboring edges have labels that are adjacent numbers (and so they have gcd 
consecutively. By maximality of the walk, every vertex in the walk (including the endpoints) is now valid; now, consider the subgraph 
.
,
.
is still connected, so by the induction hypothesis, we can label the edges of 
.
,
so labeling the remaining edges in arbitrary order will be satisfying the condition. Thus we're done.
degree.
.
.
is connected to goes from degree 1 to degree 2 when you include 
edges in this tree, we have the first 
