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.
and let 
and 
be permutations of 
. Suppose there exists a permutation 
such that 
for all 
in .
is the number of possible pairs of permutations 
, find the remainder when is divided by 1000.
and 
Prove that 
be real numbers such that 
. Find the minimum of 
and 
Prove that 
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
that satisfy the following property: if 
, 
, 
, 
are all the positive divisors of 
with 
, then 
divides 
for every 
.
be an acute-angled triangle with 
. Let 
be the circumcircle of . Let be the midpoint of the arc 
of containing 
. The perpendicular from to 
meets 
at 
and meets again at 
. The line through parallel to meets line 
at 
. Denote the circumcircle of triangle 
by 
. Let meet again at 
. Prove that the line tangent to at 
meets line on the internal angle bisector of 
.
be a positive integer. A Japanese triangle consists of 
circles arranged in an equilateral triangular shape such that for each 
, 
, 
, , the 
row contains exactly 
circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with 
, along with a ninja path in that triangle containing two red circles., find the greatest 
such that in each Japanese triangle there is a ninja path containing at least red circles.
, determine all infinite sequences of positive integers 
, 
, for which there exists a polynomial of the form ![\[ P(x)=x^k+c_{k-1}x^{k-1}+\dots + c_1 x+c_0, \]](http://latex.artofproblemsolving.com/1/9/5/195d7c158bb2b4ccf4480d5d13c22583d3f4d711.png)
where 
, 
, \dots, 
are non-negative integers, such that ![\[ P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k} \]](http://latex.artofproblemsolving.com/e/9/7/e9725833bb88b0b27fdbfef3b9111b6fe6dad372.png)
for every integer 
.
be an equilateral triangle. Let 
be interior points of such that 
, 
, 
, and
Let 
and 
meet at 
let 
and 
meet at 
and let 
and 
meet at 
is scalene, then the three circumcircles of triangles 
all pass through two common points.
medals cutoff?
and 
Prove that
