[/quote]
,
such that
non-negative real numbers,but 
0![\[1\leq \frac{a+b}{a+4b+c}+\frac{b+c}{b+4c+a}+\frac{c+a}{c+4a+b}\leq \frac{4}{3}\]](http://latex.artofproblemsolving.com/8/b/7/8b7d7b8cbcab8f2eed31e17aac717fa3c70c92f1.png)
with 
.
be orthocenter 
and 
midpoint 
.
perpendicular to 
.
be a point on the segment 
such that 
and point 
be the intersection of 
and 
.
concyclic
be an integer. Consider an 
chessboard consisting of 
unit squares. A configuration of 
rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer 
such that, for each peaceful configuration of 
square which does not contain a rook on any of its 
unit squares.
students. Let, for 
,
student have 
many friends. For 
let us define 
to be the number of students who have strictly more than 
friends. Show that 
such that:
with all 
is real number
be a line. Let 
at points 
respectively. Given that these six points on 
,
are coaxal.
,
be the least prime that does not divide 
,
to be the product of all primes less than 
For 
,
.
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
.
.
and 
Prove that
,
be the foot of the altitude from 
to 
be the reflection of 
.
intersects line 
.
Y by HWenslawski
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I have been doing one set weekly, so I think I will do ok. How about you?
Edit:41 votes!
Edit: Thank you for 80 votes on Christmas Eve!
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Edit:41 votes!
Edit: Thank you for 80 votes on Christmas Eve!
Edit: 100 VOTES!
Edit: 150 VOTES!
Edit: 200 VOTES!!
Edit: 275 VoTeS!!!
Edit: 300 VOtES!!
This post has been edited 8 times. Last edited by wisewigglyjaguar, Mar 8, 2025, 2:59 PM
it was because I was sick that day
