[/quote]
,
be a positive integer, and let 
be the set 
.
and 
be difined as:
Which is larger, 
or 
?
be a square with diagonals 
and 
,
a point in one of the sides of the square. Show that the sum of the distances from P to the diagonals is constant.
touch sides 
and 
at 
and 
respectively. Let 
and 
denote the circumcircles of triangle 
and 
respectively.
and 
intersect at 
and 
and 
intersect at 
and 
and 
and 
Prove that 
and 
Show that 
and 
are concurrent.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
such that, for all positive integers 
and 
,
,
.
,
.
,
intersects 
.
is midpoint of 
is orthocenter of triangle 
.
so that 
.
coincide
,
.
lies on segment 
such that 
.
and 
be the circumcenter and incenter of triangle 
,
and 
be the circumcenter and incenter of triangle 
,
and 
intersect at point 
.
.
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
,
satisfies 
and 
.
to line 
meets the angle bisector of 
.
holds.
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
be a positive integer divisible by 3, and let 
be non-negative real numbers satisfying 
.![\[
M_n = \min_{1 \le i \le n} \{x_i x_{3i}\},
\]](http://latex.artofproblemsolving.com/d/b/b/dbbf0f6bf20f506d6cb63357b06ec6c52bebed10.png)
where the indices are taken modulo 
.
with integer coefficients which satisfy the following conditions:
is a positive integer for any positive integer 
for all positive integers 
and 
and 
are on the same side of 
be intersection of 
with 
and 
be a function satisfying the following conditions:
,
and 
,
;
is surjective.
.
,
.
,
,
,
Y by Crazy_LittleBoy, Grotex, Havister, Adventure10, Modern_Hunter, IraeVid13, Mango247, SunnyEvan, MS_asdfgzxcvb, and 2 other users
Let 
and 
are two sets of nonnegative quantities. Now, if we have all the following conditions fulfilled:
(generally the
-th elementary symmetric functions 
for any natural number with 
)
then
for 
.
and 
are two sets of nonnegative quantities. Now, if we have all the following conditions fulfilled:
(generally the
-th elementary symmetric functions 
for any natural number with 
)then
for 
.
