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 the orthocenter of an acute triangle 
. Consider the circumcenters of triangles 
, 
, and 
. Prove that they are the vertices of a triangle that is congruent to .
denote the set of all real numbers. Find all functions 
such that
for all real numbers 
, 
.
and 
Prove that
Let and 
Prove that
Let and 
Prove that
be a convex quadrilateral with 
, 
and 
. Denote by 
and 
the reflections of 
in lines 
and 
, respectively. Suppose that the segments 
and 
meet the line 
at 
and 
, respectively. Prove that the circumcircles of triangles 
and 
are tangent to each other.
on 
without derivative.
be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer 
the numbers 
leave different remainders upon division by ..
![$\begin{cases} x^3+x^2-xy^2=\sqrt{(x-y^2)^3} \\ 56x^2+20(x^2-y^2)=\sqrt[3]{4x(8x+1)}-2\end{cases}$](http://latex.artofproblemsolving.com/2/4/7/2472f62c2e2e6c684bc262e61da80cba4d0d263c.png)
there exists positive integers 
such that
be a scalene triangle with incircle 
. Denote by 
the midpoint of arc 
in the circumcircle of , and by 
the point where the -excircle touches . Suppose the circumcircle of 
meets again at 
and intersects at two points 
, 
.
or 
is tangent to .
be positive real numbers such that 
![\[\sqrt{2014a^2+7}+\sqrt{2014b^2+7}+\sqrt{2014c^2+7}\leq \sqrt{2014+7}\ (a+b+c).\]](http://latex.artofproblemsolving.com/d/d/d/ddd3373f70e94312e7488765cdb3e5f070ac1cc2.png)
be a 
-element subset of the set 
. Prove that there exist numbers 
, 
in 
such that the sets ![\[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \]](http://latex.artofproblemsolving.com/d/5/c/d5c2a7b943e8c7200361698f3eb6f83700ad68d6.png)
are pairwise disjoint.
on the board. Every minute Ivan makes next operation: takes any number 
written on the board, except ( Can be same numbers on the board). After some time on the board there are 
numbers. is it possible?
be real numbers satisfying 
Prove that 
and determine all cases of equality.
, 
, segment 
. Now we have a 
which is similiar to (the vertexs of them might not correspond). Find the maximum value of 
.
get any larger than 
or 
? I don't see why there is a upper bound.
get any larger than or ? I don't see why there is a upper bound.
to be distinct (and lie in the same plane).
or 
.