[/quote]
,
such that for every positive real 
,
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
,
.
, 
then prove that 
>
are non-negative real numbers. Prove that![\[( {a}^{2}-bc ) \sqrt {a+b+c}+ ( {b}^{2}-cd) \sqrt {b+c+d}+ ( {c}^{2}-da ) \sqrt {c+d+a}+ ( {d}^{2}-ab ) \sqrt {d+a+b} \geqslant 0.\]](http://latex.artofproblemsolving.com/8/2/6/826e721b038794ce0cd6ca01b7afb5567a3efffe.png)
are non-negative real numbers and 
![\[\frac{a(a^2-bc)}{\sqrt{ka+b+c}} + \frac{b(b^2-ca)}{\sqrt{kb+c+a}} + \frac{c(c^2-ab)}{\sqrt{kc+a+b}} \geqslant 0.\]](http://latex.artofproblemsolving.com/5/f/9/5f9deb36a5aee41d3fa177b53391b7149fb76a6e.png)
(ii) Prove that![\[(a^2-bc)\sqrt{ka+b+c}+(b^2-ca)\sqrt{kb+c+a}+(c^2-ab)\sqrt{kc+a+b} \geqslant 0.\]](http://latex.artofproblemsolving.com/2/c/4/2c4e7a84750ebc1521a0190076b116ada67e3937.png)
be a point in the plane, and 
a line not passing through 
satisfying
Prove
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
.
matrix such that 
,
is a permutation of 
,
.
is a permutation of 
.
Y by GeoKing, Rounak_iitr
Let 
and 
be the circumcenter and orthocenter of a scalene triangle 
, respectively. Let 
be the intersection point of the lines 
and 
. Suppose the line 
meets the side at 
. Let 
and 
be the second intersection points of the circumcircles of 
and 
with the circumcircle of 
, respectively. Show that the four points 
and lie on a circle.
and 
be the circumcenter and orthocenter of a scalene triangle 
, respectively. Let 
be the intersection point of the lines 
and 
. Suppose the line 
meets the side at 
. Let 
and 
be the second intersection points of the circumcircles of 
and 
with the circumcircle of 
, respectively. Show that the four points 
and lie on a circle.This post has been edited 1 time. Last edited by a_507_bc, May 3, 2024, 10:18 AM
and 
Also, let 
be the midpoint of 
and let 
be the midpoint of 
Note that 
is the nine-point circle of 
which we will denote with 
are collinear.
be the antipode of 
.
are collinear, so if 
intersects 
for a second time at point 
,
However, we know that 
so 
and we are done.
are collinear.
centered at 
Note that under this inversion, we have that 
and 
,
Hence, 
and 
.
Since 
,
Note that 
and in particular 
Therefore, 
is cyclic iff 
is cyclic. However, points 
all lie on the circle with diameter 
and we may conclude.
be the orthic triangle, let 
where 
are midpoints of 
,
inversion the queue points go to 
are collinear but this follows as 
are antipodes of 
and 
.
be circle with diameter 
other then 
we have 
cyclic.
inversion about 
, 
and 
.
.
are cyclic 
cyclic
Let 
be the antipodes of 
on 
be the foot of the altitude from 
to 
.
is cyclic.![\[\frac{r+a+b+c}{2}=a\iff r=a-b-c\]](http://latex.artofproblemsolving.com/c/0/5/c051a406274e309a65d0de4a89bb599563e6f8df.png)
![\[s=\frac{\overline{(a+b+c)}(a-b-c)+(a+b+c)\overline{(a-b-c)}}{2\overline{(a+b+c)}}=\frac{(ab+bc+ca)(a-b-c)+(a+b+c)(bc-ab-ac)}{2(ab+bc+ca)}\]](http://latex.artofproblemsolving.com/3/8/3/383e4c5c5a7ca65986086aba5a692fb8272bc82f.png)
![\[s=\frac{-ab^2-abc-ac^2}{(ab+bc+ca)}\]](http://latex.artofproblemsolving.com/a/f/6/af66ef05323a993327c979578b110720881e2403.png)
![\[\frac{k-r}{k-s}.\frac{l-s}{l-r}=\frac{c-a}{-b+\frac{ab^2+abc+ac^2}{(ab+bc+ca)}}.\frac{-c+\frac{ab^2+abc+ac^2}{(ab+bc+ca)}}{b-a}\]](http://latex.artofproblemsolving.com/c/1/4/c146933f20d5e61696444ac8f8da2c3de745e0c3.png)
![\[=\frac{c-a}{b-a}.\frac{ab^2-bc^2}{ac^2-b^2c}=\frac{b(c-a)(ab-c^2)}{c(b-a)(ac-b^2)}\]](http://latex.artofproblemsolving.com/4/c/b/4cb2084c042395f4ce7ba514fef4f67c30a41312.png)
![\[\frac{b(c-a)(ab-c^2)}{c(b-a)(ac-b^2)}\overset{?}{=}\overline{(\frac{b(c-a)(ab-c^2)}{c(b-a)(ac-b^2)})}=\frac{(a-c)(c^2-ab)b}{(a-b)(b^2-ac)c}\]](http://latex.artofproblemsolving.com/a/c/d/acd7c2dba9886660414589d983fadc86dd17e313.png)
Which is true as desired
be the orthic triangle of 
.
,
and 
.
and 
are the midpoints of 
and 
is 
,
lies on line 
.
,
:
;
.
is cyclic which means 
.
is circle with diameter 
means 
at 
and 
be the point 
.
be a point on the line 
so that 
.
.
