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.
-tuple 
, 
, of symbols from 
, we define a sequence 
according to the following rule: If 
, then 
, where 
if 
(taking 
) and 
is the symbol other than 
if 
. Find all positive integers 
for which there exists some integer 
such that 
.
be an acute triangle with orthocenter 
. Let 
, 
and 
be the feet of the altitudes of triangle opposite to vertices 
, 
, and 
respectively. Let 
and 
be the midpoints of 
and 
, respectively. Let 
be the intersection of lines 
and 
. Prove that is the circumcenter of triangle if and only if is the midpoint of 
.
and 
be two prime numbers, with being a divisor of 
. Prove that there exist integers 
such that the polynomial 
is divisible by the polynomial 
with 
is a multiple of and 
.
be the circumcircle of acute triangle . Points 
and 
are on segments 
and 
respectively such that 
. The perpendicular bisectors of 
and 
intersect minor arcs and of at points 
and 
respectively. Prove that lines 
and 
are either parallel or they are the same line. be a scalene acute triangle with incenter 
and circumcircle 
. 
is the midpoint of small arc 
on and 
is the projection of onto the line passing through the midpoints of and . A circle 
with center 
is internally tangent to at , and touches segment . If the circle with diameter 
meets again at 
, prove that 
bisects 
. be circumscribed about circle 
, and let be the orthocenter of 
. The circle touches line at . The tangent to the circle 
at meets at 
. Let be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
be two nonzero real numbers such that 
Consider the functions 
and 
satisfying the condition ![$$g(f(x+by))=a[f(x)+2yg(x)]+2xg(y)+bg(y)[y+g(y)],\quad\forall x,y\in\mathbb{R}.$$](http://latex.artofproblemsolving.com/3/c/9/3c974cb76aae7f529dcd85202b0c7d441a07d002.png)
(a) Prove that there exists a real number such that 
(b) Find all functions 
and 
that satisfy the given condition.
be a square and a point lies on the interior of such that triangle 
is equilateral. Evaluate 
to 
(inclusive) are selected randomly without repetitions, then either all of them will be odd, or all will be divisible by 
, or all will be divisible by 
with 
is coprime to 
. It is known that for every 
, there always exist 
satisfying 
Prove that is a cube.
and 
such that![\[
1^n + 2^n + 3^n + \cdots + n^n = x!
\]](http://latex.artofproblemsolving.com/6/4/b/64b5f5919fd3710cc15e96724918eae216c8e4f0.png)
crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least 
crosses. for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors.
and 
Prove that
, such that 
Prove that :
and 
are given that are, respectively, the circumcenter of a triangle 
perpendicular to 
at 
and the circle 
that inersects 
.
intersects 
at 