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 acute triangle with 
. 
are the orthocenter and excenter. Let 
be a point on the same side of 
as 
, such that 
is equilateral. Let 
be a point on the same side of 
as , such that 
is equilateral. Let 
be a point on the same side of 
as , such that 
is equilateral. Let 
be the midpoint of 
. Prove that 
are collinear.
be real numbers satisfying 
and 
Prove that 
Let be real numbers satisfying 
and 
Prove that 
and a line 
in the plane.Let 
be reflections of 
across the line , respectively.Let 
be the midpoints of 
, respectively.Let 
be the reflections of across , respectively.Prove that the points lie on a line parallel to .
and 
are natural numbers such that 
is divisible by 
and 
is divisible by 
, then find the least possible value of 
.
be an interior point of the acute triangle with 
so that 
The point 
on the segment 
satisfies 
the point 
on the segment 
satisfies 
and the point 
on the line satisfies 
Let 
and 
be the circumcenters of the triangles 
and 
respectively. Prove that the lines 
and 
are concurrent.
be a positive integer and let 
, 
,..., 
be positive integers from set 
such that every number from this set occurs exactly once. Is it possible that numbers , 
all have different remainders upon division by , if:
such that for all 
,![\[
mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).
\]](http://latex.artofproblemsolving.com/c/0/c/c0cdf1bcb8053f5dc26e9df6355207b5e479f861.png)
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
be a scalene triangle with circumcircle 
and incenter 
. Ray 
meets 
at and meets again at ; the circle with diameter 
cuts again at 
. Lines 
and 
meet at 
, and 
is the midpoint of 
. The circumcircles of 
and 
intersect at points 
and 
. Prove that passes through the midpoint of either 
or 
.-by- board of unit squares for some odd positive integer . We say that a collection 
of identical dominoes is a maximal grid-aligned configuration on the board if consists of 
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let 
be the number of distinct maximal grid-aligned configurations obtainable from by repeatedly sliding dominoes. Find the maximum value of as a function of .
that satisfy the equation
.
