AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a convex cyclic quadrilateral with circumcircle 
. Let 
produced beyond 
meet 
produced beyond 
, at 
. Let 
be a line through meeting 
and 
at 
and 
respectively, so that 
(respectively 
) are on opposite sides of (resp. 
). Suppose 
and 
are points on the arc 
of not containing 
so that 
are tangent to . Prove that 
are collinear.
, 
, 
be real positive numbers. Show that ![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]](http://latex.artofproblemsolving.com/0/9/5/0959464bbb9b821a11df292f9e6addded7326371.png)
is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
the ratio 
is an integer.
+1 w
be a natural number. At a mathematical olympiad training camp the same courses are organised every day. Each student takes exactly one of the courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of , the largest possible number of students at the camp?
such that 
for all 
?
have the same incenter 
.Prove that 
is concyclic iff 
is concurrent
and be positive integers. Suppose that the sum of integers between and , including and , are equal to 
.
, find the pair such that achieves the minimum.
be real numbers such that 
and
and 
denote two sequences of integers defined as follows:
Prove that, except for the "1", there is no term which occurs in both sequences.
and the same for 
then,
![\[x_n=X(-1)^n+Y2^n.\]](http://latex.artofproblemsolving.com/b/6/4/b64103ce6b36840afb72e204ce9d8bde7c3ae59e.png)
Using 
and 
yields ![\[x_n=\frac{2^n+(-1)^{n+1}}{3}.\]](http://latex.artofproblemsolving.com/0/e/9/0e91ecb368a2adce3368cc089356ac41c89c7f0a.png)
Similarly, solving for the second recurrence yields ![\[y_n=2\cdot 3^{n-1}+(-1)^n.\]](http://latex.artofproblemsolving.com/b/e/9/be9151fca1e0c704e75c410c3d6963099b9170e1.png)
So if 
then 
or 
.
or 
, then 
is the only solution, corresponding the the fact that the term 
is in both sequences. If 
, then 
. But we have 
, so ![\[2\cdot 3^n+3(-1)^n+(-1)^m\equiv 5(-1)^n+(-1)^m\pmod 8\]](http://latex.artofproblemsolving.com/c/9/6/c965e92b534b4e7746ecd90274be64ceaa3d0122.png)
which cannot be 
. Hence there are no solutions for , and the only integer in both sequences is indeed . 
cannot be equal to 
. 
.
. Now suppose this is true for some 
. Since 
, and 
, so adding these two gives 
, which means will always be greater than for all . 
.
Where 
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
and the same for 
then we would have 
.
,
and 
,
,
.
.
,
,
or 
,
.
.
.
if 
,
.
where 
and 
are roots of the equation 
.
and 
we get 
.
.
we get 
.
if 
, so we get 
.
we get 
which is not true hence , sequence 
and 
shares no common numbers other than 