[/quote]
,
satisfy that: 
such that![\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]](http://latex.artofproblemsolving.com/e/b/c/ebccae7e765d9fc3c8605e58a7399a70127b39b1.png)
and natural number 
,
, 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 
and 
Prove that 
and 
intersect on the circumcircle of 
.
grid with 
rows and 
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 Mango247
The 
-excircle of the triangle 
touches the extension of the side 
at the point 
. Let 
be the incenter of the triangle, and 
be the midpoint of the segment 
. Prove that the points 
and 
lie on the same circle.
(Anton Trygub)
-excircle of the triangle 
touches the extension of the side 
at the point 
. Let 
be the incenter of the triangle, and 
be the midpoint of the segment 
. Prove that the points 
and 
lie on the same circle.(Anton Trygub)
This post has been edited 1 time. Last edited by parmenides51, Jun 6, 2021, 8:16 AM
