[/quote]
,
, 
then prove that 
>
such that for every positive real 
,
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
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 
.
be an acute, non-isosceles triangle, 
be its heights and 
its circumcircle. 
cuts the circumcircle at points 
,
being between points 
.
be the midpoints of the major and the minor arc 
,
cuts 
.
and 
cut again 
at points 
and 
,
is the line passing through the circumcenters of triangles 
and 
the parallel lines through 
to 
meets 
at 
and 
meets 
at 
,
are concyclic.
Y by
Points 
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
N. Sheshko, D. Zmiaikou
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
N. Sheshko, D. Zmiaikou
This post has been edited 1 time. Last edited by nAalniaOMliO, Oct 31, 2024, 10:12 AM
)
and say it's rational coordinates are (p,q) then move the point to ([p],[q]) where [•] denote the box function and we are done with all points with integral coordinates.
and 
is clearly 1, where the first point is a lattice point and the second one has rational coordinates.
