[/quote]
,
,
be all the positive integers smaller than 
such that
for all 
.
such that the following inequality holds for all 
:![\[
ab + bc + ca \leq \frac{(a + b + c)^2}{3} + k \cdot \max \{ (a - b)^2, (b - c)^2, (c - a)^2 \} \leq a^2 + b^2 + c^2
\]](http://latex.artofproblemsolving.com/1/c/e/1ce66163068b499383adb4a4ecec86c5f4edaec3.png)
such that 
we have![\[\max(|\sin x|,|\sin (x+1)|)>\frac13.\]](http://latex.artofproblemsolving.com/e/d/e/ede9b524370c457fe41038cca5663e3c9bdaf320.png)
,
lie on a line in this order and satisfy 
.
and 
and 
,
is acute, and let 
be its orthocentre. Points 
lie on lines 
and 
,
and 
and 
and 
of positive integers is called central if for every positive integer 
,
terms of the sequence is equal to 
of positive integers such that for every central sequence 
,
.
be all the positive integers smaller than ![\[gcd(N, c_i + c_{i + 1}) \neq 1\]](http://latex.artofproblemsolving.com/1/0/7/1075290d442e8c1f54237aad8ad07126f455f1f1.png)
for all 
.
is extended beyond 
so that 
.
,
,
,
be positive real numbers such that 
.![$$\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+a^3}{2}}\leqslant a^2+b^2+c^2.$$](http://latex.artofproblemsolving.com/5/a/b/5ab64bd6c09363206f936a4a49a9ac7968218600.png)
satisfied 
Find minimum and maximum of 
of positive integers is called central if for every positive integer 
,
,
,
of positive integers such that for every central sequence 
,
,
,
and 
and 
.![\[
\frac{1}{x} + \frac{1}{y} + 2 \sqrt{\frac{2}{x^2 + y^2}} + 4 \geq 4 \left( \sqrt{\frac{2}{x^2 + 1}} + \sqrt{\frac{2}{y^2 + 1}} \right).
\]](http://latex.artofproblemsolving.com/1/b/8/1b8ab2e343a3043271f5a392d32b12aca748f1b9.png)
Y by CZRorz, tenplusten, HoRI_DA_GRe8, samrocksnature, Adventure10
Let there be a scalene triangle 
, and its incircle hits 
at 
. The perpendicular bisector of 
meets the circumcircle of at 
, where 
is on the same side with 
with respect to . Let the line parallel to 
and passing through 
meet 
at 
. Prove that the intersection between and 
lies on the circumcircle of 
.
, and its incircle hits 
at 
. The perpendicular bisector of 
meets the circumcircle of at 
, where 
is on the same side with 
with respect to . Let the line parallel to 
and passing through 
meet 
at 
. Prove that the intersection between and 
lies on the circumcircle of 
.
.
are colinear where 
is the incenter. This implies that 
and 
is cyclic.
.
and it suffices to show that 
.
be the orthic triangle of 
.
(Iran) and quadrilaterals 
and 
are cyclic. Hence 
as desired.
,
. Note that 
,
hence
as desired.
.
to be the external bisector of 
;
.
is cyclic where 
we want to prove that 
Then by Menelaus on 
and line 
,
again at 
.
is perpendicular 
which implies 
.
.
and since 
hence, 
.
.
by 
.
.
of 
.
.
.
is the internal angle bisector of 
is the mid-point of arc 
.
is the antipode of 
are concurrent at the Gregonne Point of 
.
.
also notice that 
.
where 
.
And it's about 15-20 minutes left to 
, so I hurry up 
, so 
, thus 
is cyclic .
and 
and 
the intersection of 
and 
respectively. Notice that 
so 
and 
is cyclic. By the Midpoint Lengths Lemma and Power of a Point, 
and 
is cyclic. 
