[/quote]
,
(use in a problem for junior) to practice. Can anyone share to me?
be a square with 
side length and with center 
.
is drawn where point 
lies on line 
.
be an odd integer, 
.
integers 
such that for any 
,
and 
.
for which satisfy 
for some 
,
is greater than 
.
such that![\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]](http://latex.artofproblemsolving.com/e/b/c/ebccae7e765d9fc3c8605e58a7399a70127b39b1.png)
, such that 
Prove that :
equality cases : ?
distinct positive integers such that adding the product of any two of them to 
yields a perfect square?
be a sequence of positive integers with 
,
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
.
is a prime less than 
for some positive integer 
,
such that 
.
for which
be an acute angled scalene triangle with circumcircle 
.
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
be points on segments 
Prove that 
and 
intersect on the circumcircle of 
.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of 
Y by blacksheep2003
Let 
be a non isosceles triangle with circumcircle 
and incircle 
. Denote 
as the circle that external tangent to at 
and also tangent to the lines 
at 
respectively. Define the circles 
and the points 
similarly.
1. Denote J as the radical center of
and suppose that 
intersects at the second point 
intersects 
at the second point Y , JC' intersects 
at the second point 
. Prove that the circle 
is tangent to 
.
2. Prove that
are concurrent at the point 
and 
points 
are collinear.
be a non isosceles triangle with circumcircle 
and incircle 
. Denote 
as the circle that external tangent to at 
and also tangent to the lines 
at 
respectively. Define the circles 
and the points 
similarly.1. Denote J as the radical center of
and suppose that 
intersects at the second point 
intersects 
at the second point Y , JC' intersects 
at the second point 
. Prove that the circle 
is tangent to 
.2. Prove that
are concurrent at the point 
and 
points 
are collinear.
hence 
is the radical center of 
so we have that
then we have
hence 
is tagent to 
as well(complete the proof).
be the center of the excircle WRT 
of 
,
tagents to 
at 
respectively,we known that 
are concurrent(prove this by Ceva's theorem)(1)
hence the image of the mixtilinear cirlce WRT the vertex 
of 
which implies that 
are isogonal WRT 
are isogonal WRT 
and 
are isogonal WRT 
(
are concurrent at 
is the midpoint of ar
,
similarly with 
hence 
.
and 
so we have that 
(complete the proof).
)
with 
,
,
and 
.
.
,
,
.
and 
,
