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 triangle and point 
be 
altitude of 
.
be a point on 
and 
such that 
and 
. Point 
is the intersection of 
and 
. Point 
be intersection of 
and 
. Line 
intersect circumcircle of again at 
.
.
bisect .
Define a cell with 
property if after remove that cell from the board The board can be tile with 
tetromino. cell \newline Note: tetromino can be rotated but not flipped
be a positive integer such that 
.
.
such that 
for all real number 
and 
be a scalene triangle with point and on the plane such that 
is an equilateral . Let intersect 
and 
at 
and 
respectively and intersect 
and 
at 
and 
respectively .
and 
.
?
such that 
, and consider all subsets of elements of the set 
. Each subset has a smallest element. Let 
be the arithmetic mean of these smallest elements. Prove that: ![\[ F(n,r)={n+1\over r+1}. \]](http://latex.artofproblemsolving.com/e/9/d/e9da0a074ee712e67c3d999193a933bded5f98f5.png)
satisfying
be the positive real number. Prove that
and 
be the circumcenter and orthocenter of an acute angled triangle 
Let be the midpoint of 
Let 
be the point on the angle bisector of 
such that 
Let 
be the point such that 
is a rectangle. Prove that 
are collinear.
be a prime number. Prove that there exists a natural number 
such that![\[
p \mid 2024^n - n.
\]](http://latex.artofproblemsolving.com/1/4/a/14aea55ed2bcea62d2a0d8d0770513715d8c4fb8.png)
be positive real numbers . Prove that
.
, which implies that 
(since 
), or c=0,9.
since 
.
work.
, which is a contradiction since 
.
.
whereas 
(this occurs when a,k are both minimal at 1), whence there are no solutions for this case.
, i.e. m|n if n is a multiple of 1000.
Prove that
Prove that
be a five digit number (whose first digit is non-zero) and let 
be the four digit number formed from 
is an integer.
.
obviously has the same first two digits as 
.
.
,
,
.
,
,
.
gives 
,
.