[/quote]
,
,
,
,
be distinct positive integers, 
.
and 
such that 
does not divide any of the numbers 
,
,
.
board, 
pieces are placed (vertically or horizontally) so that they occupy entirely 
boxes on the board and do not have a common point.
,
be the sum of the digits of 
such that there are infinite positive integers 
.
,
be fixed positive integers. There are 
ducks sitting in a
be the circumcenter of triangle 
.
intersects the altitude from 
at point 
.
be the midpoints of 
,
respectively. If 
intersects 
at 
,
meets 
at 
,
is cyclic.
![\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]](http://latex.artofproblemsolving.com/2/a/7/2a73a5e58eab4a994fbd83160fb79a0b00152951.png)
and circumcentre 
be the touchpoints of the incircle with 
,
is the Euler line of 
.
with its incircle 
.
and 
be the points of tangency of 
and 
,
and 
such that 
is tangent to 
is the intersection point of the lines 
and 
.
and 
are parallel or coincide.
and let 
be the midpoint of 
Suppose that 
different from 
such that 
Prove that the orthocenter of 
lies on the circumcircle of 
.
that satisfy
for all integers 
and rational numbers 
.
are one of such 
Y by Adventure10
Circles 
are given in a plane. 
and 
touch each other externally, and both touch 
internally at 
and 
respectively. The common internal tangent to and meets at 
and 
Let 
and 
be the intersections of 
and 
with and , respectively. Prove that 
is a common tangent to 
are given in a plane. 
and 
touch each other externally, and both touch 
internally at 
and 
respectively. The common internal tangent to and meets at 
and 
Let 
and 
be the intersections of 
and 
with and , respectively. Prove that 
is a common tangent to 
This post has been edited 1 time. Last edited by IstekOlympiadTeam, Nov 28, 2015, 8:24 AM
Reason: qrong source
Reason: qrong source
and line 
where 
will be the intersection of 
will be the point on 
concyclic
perpendicular with parallel lines
are perpendicular lines too.
tangent with 
(We can do similar way with other side)
(Because 
(Because 
)
,
is tangent to both lines, which is a good staring exercise for the reader.
with center 
and 
and 
and 
is tangent to 
and 
is tangent to 
