[/quote]
,
positive real numbers such that 
. Prove that :
such that for all 
,![\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]](http://latex.artofproblemsolving.com/d/1/0/d10c01d7d416243614b24f04218d157e8b025ff2.png)
.
to 
meets 
at points 
,
to 
is 
.
intersects line 
,
intersects line 
,
intersect at 
.
is tangent to 
.
be an acute triangle. Points 
,
.
and 
and 
,
is acute, and let 
be its orthocentre. Points 
lie on lines 
and 
,
and 
and 
and 
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers 
and 
from the board such that:![\[
a^2 + b^2 + 1 = 2ab
\]](http://latex.artofproblemsolving.com/8/f/c/8fc5df8dbc8e1647edfaa96dcfa8a0dfdac0c961.png)
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers 
and 
from the board such that:![\[
c^2 + d^2 + 2025 = 2cd
\]](http://latex.artofproblemsolving.com/2/7/0/2706d72797c5e392b3f068a8174bd257e4d93fa1.png)
,
are even?
and 
be real numbers such that 
.
and 
.
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
be an acute-angled, non-isosceles triangle with circumcenter 
and incenter 
,![\[
\prod_{\text{cyc}} \left( \frac{1}{a+b-c} + \frac{1}{a+c-b} - \frac{2}{b+c-a} \right) \neq 0,
\]](http://latex.artofproblemsolving.com/7/8/1/78125759bf5c73149e4a56538e9b0c44309704a7.png)
,
,
.
,
,
be the excenters opposite to vertices 
,
,
,
,
,
be the Lemoine points of triangles 
,
,
,
,
,
all pass through a common point 
.
.
intersect at point 
.
and 
with their common external tangents lie on a circle 
.
and 
with their common external tangents lie on a circle 
.
Y by Adventure10, Mango247
Let 
be a diameter of the circunference 
, let 
and 
be point in this circunference, such that 
is perpedicular to . Let 
be the point of intersection of the segment and the segment , and a point 
that is in the segment 
is different of . The lines 
and 
intersects , in 
and 
respectively. If 
is the circumcenter of triangle 
, show that the area of triangle 
is invariant, independent of the position of the point .
be a diameter of the circunference 
, let 
and 
be point in this circunference, such that 
is perpedicular to . Let 
be the point of intersection of the segment and the segment , and a point 
that is in the segment 
is different of . The lines 
and 
intersects , in 
and 
respectively. If 
is the circumcenter of triangle 
, show that the area of triangle 
is invariant, independent of the position of the point .This post has been edited 2 times. Last edited by mathisreal, Mar 24, 2019, 10:31 PM
be the center of the circle 
,
and 
.
,
.
,
.
is 
.
.
and does not depends on 
