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.
is divided into cyclic quadrilaterals with pairwise disjoint interiors. None of the vertices of the cyclic quadrilaterals in the decomposition is an interior point of a side of any cyclic quadrilateral in the decomposition or of a side of the quadrilateral . Prove that is also a cyclic quadrilateral.
and a positive integer 
, we denote the -fold composition of by 
. A polynomial with real coefficients is called perfect if for each integer 
there is a positive integer so that 
is an integer. Is it true that for each perfect polynomial , there exists a positive 
so that for each integer there is 
for which is an integer?
for which there exists a positive integer 
with the property that is divisible by and 
is divisible by 
.
be a positive integer, with divisors 
. Prove that 
is always less than 
, and determine when it is a divisor of .
and 
be sequences of real numbers for which 
and
for all positive integers . Prove that is eventually increasing (that is, there exists a positive integer 
for which 
for all 
).
be an integer. Let 
, 
, ..., 
be positive real numbers such that ![\[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\]](http://latex.artofproblemsolving.com/1/a/b/1ab410ebc17d9f6c38a3e7ae0f6efa955ffddfd3.png)
Show that 
, 
, 
are side lengths of a triangle for all 
, 
, with 
.
be an acute triangle with incenter 
and circumcenter 
. The incircle touches sides 
and 
at 
and 
respectively, and 
is the reflection of 
over . The circumcircles of 
and 
meet at 
, and the circumcircles of 
and meet at a point 
, where 
is the midpoint of 
. Prove that if 
and meet at 
, then 
.
and 
Prove that
such that divides 
but 
does not divide 
.
with right angles at and 
has an inscribed circle with center and radius 
. Let the lengths of the parallel sides 
and 
be 
and 
, and the intersection of the diagonals 
and 
be 
. to one of the trapezoid sides has the length . and as a function of and .
, construct a point inside the quadrilateral such that triangles 
and 
are similar. Prove that triangle 
has sides 
proportional to 
, 
, and 
, respectively.
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
