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 triangle. The incircle of touches the sides 
and 
at the points 
and 
, respectively. Let 
be the point where the lines 
and 
meet, and let 
and 
be points such that the two quadrilaterals 
and 
are parallelogram.
.
be an acute triangle with incenter 
, circumcenter 
, and circumcircle 
. Let 
be the midpoint of 
. Ray 
meets 
at 
. Denote by 
and 
the circumcircles of 
and 
, respectively. Line 
meets at 
and , while line 
meets at 
and 
. Assume that lies inside 
and 
. at and and the tangents to at 
and . Given that 
, prove that these four lines are concurrent on .
and 
be positive real numbers. Prove that
find, in terms of , the minimum value of 
for which there is a set of 
distinct positive integers that has sum greater than but every subset of size has sum at most 
is called lucky if it has at least two distinct prime divisors and can be written in the form:![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
where 
are distinct prime numbers that divide . (Note: it is possible that has other prime divisors not among .) Prove that for every prime number 
, there exists a lucky number such that 
.
be a sequence of positive real numbers such that for every 
, we have:![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
, 
, , 
, where and 
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
, such that for any positive integer 
and non-negative reals 
, the following inequality always holds:
Here 
and 
denote the median and arithmetic mean of 
, respectively.
.
and 
Prove that
such that 
.![$$2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$$](http://latex.artofproblemsolving.com/f/0/7/f0731fbe5c83aedc8474975f930f7194c08cc070.png)
Prove that
Where 
,such that 
P
