[/quote]
,
such that![\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]](http://latex.artofproblemsolving.com/e/b/c/ebccae7e765d9fc3c8605e58a7399a70127b39b1.png)
(use in a problem for junior) to practice. Can anyone share to me?
, 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 
.
,
.
,
,
with 
,
,
,
and 
intersect at 
.
and 
intersect again at 
.
and 
intersect the circumcircle of triangle 
again at 
and 
,
lies on 
.
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 
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 
be a square with 
side length and with center 
.
is drawn where point 
.
Y by
Let 
be an acute-angled triangle in which 
. On the side 
mark a point 
such that 
, and on the side 
mark a point 
such that the segment 
passes through the orthocenter of triangle . Prove that the center of the circumcircle of triangle 
lies on the segment 
.
be an acute-angled triangle in which 
. On the side 
mark a point 
such that 
, and on the side 
mark a point 
such that the segment 
passes through the orthocenter of triangle . Prove that the center of the circumcircle of triangle 
lies on the segment 
.This post has been edited 2 times. Last edited by parmenides51, Apr 29, 2021, 12:32 PM
