[/quote]
,
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
satsfying 
and:![\[ f(2n)f(2n+1)=9f(n)^2+3f(n)\]](http://latex.artofproblemsolving.com/b/c/1/bc18d6ad72589f012dcab7836228b3ed7ee061c2.png)
be positive integer.Prove the following equality.![\[ \sum_{r=1}^n (-1)^r\ _n C_r\ \frac{1}{r}=-\sum_{r=1}^n \frac{1}{r} \]](http://latex.artofproblemsolving.com/4/0/2/40263db7dae2474de3540974d1d3246590367983.png)
Where 
is quadratic mean 
,
is harmonic mean 
,
is arithmetic mean 
,
is geometric mean ![$\sqrt[n]{a_1a_2\cdots a_n}$](http://latex.artofproblemsolving.com/5/5/6/556f366fbe0e31be453f8b9c4ea6da4572cc66b4.png)
.
be a trapezoid inscribed in circle 
,
.
is the symmetric point of 
across 
intersects 
.
is midpoint of 
is orthocenter of triangle 
take a point 
.
coincide
.
.
such that 
.
and 
be the projections of 
.
meets again 
.
is the point of intersection of 
and 
intersect on the line 
.
by adding every 
?
on 
on 
be such that 
and 
.
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
are concurrent.
Prove that![$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$](http://latex.artofproblemsolving.com/8/b/e/8bed6605f3988f59b8395b091b92a8b9e8b40419.png)
Where 
.
intersect at 
be the line through 
the circumcircle of triangle 
.
.
have a common point.
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
we define:
A 
between 
,
,
and 
for every 
;
A 
between 
A 
between 
Y by son7, JackMathReacher, Mango247, Mango247, Mango247
At the altitude 
of triangle 
, some point 
is marked. Line 
intersects side 
at point 
, line 
intersects side 
at point 
. It is known that divides the segment 
in the ratio 
, counting from point . Find the ratio 
.
of triangle 
, some point 
is marked. Line 
intersects side 
at point 
, line 
intersects side 
at point 
. It is known that divides the segment 
in the ratio 
, counting from point . Find the ratio 
.This post has been edited 1 time. Last edited by parmenides51, Feb 5, 2021, 8:11 PM
