[/quote]
,
.
be a point on the perpendicular bisector of 
,
be the intersections of the perpendicular bisectors of 
and 
.
and 
and the point 
is given. Let 
be set of all 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
,
and 
are integers, 
.
is said to be valuable if the intersection of any two segments from 
rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to 
satisfying 
,![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](http://latex.artofproblemsolving.com/f/e/8/fe85cf52b883c214806ab837a2eb25cebb511d7b.png)
such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
(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 
.
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 
.
Y by jhu08
Let 
be a scalene acute triangle, and let 
be the midpoint of 
. Let 
and 
be the feet of the altitudes from 
and 
respectively. Let 
and 
be the midpoints of 
and 
respectively. Line 
intersects lines 
and 
at points 
and 
respectively. Prove that quadrilateral 
is cyclic.
be a scalene acute triangle, and let 
be the midpoint of 
. Let 
and 
be the feet of the altitudes from 
and 
respectively. Let 
and 
be the midpoints of 
and 
respectively. Line 
intersects lines 
and 
at points 
and 
respectively. Prove that quadrilateral 
is cyclic.This post has been edited 3 times. Last edited by parmenides51, Dec 8, 2021, 6:42 PM
Reason: source
Reason: source
be the circumcircle of 
.
be the circle centered at 
.
,
is the radical axis of 
and 
.
,
.
be 
,
and 
,
and 
,
denote the 
.
are tangent to 
are inverses under 
-
.
are symmetric about 
follows from isogonality, we know 
.
where the last equality follows from 
Thus, the homothety taking 
also maps 
is cyclic, we know 
