prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
satisfying 
is inscribed in circle 
.
denote its incenter and 
its 
-
denote the midpoint of arc 
.
meets 
.
at 
and 
at 
and 
respectively. The circumcircle of 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
that satisfy 
with 
,
and area 
.
with diameter 
at 
,
and extend to meet 
and 
is the diameter of 
of 
,
of 
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
.
lie on 
respectively, such that 
.
to meet the circumcircle of 
respectively. Let the circumcircles of 
meet at points 
.
to 
,
,![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
.
with 
,
to meet the circumcircle of 
at 
,
.
Y by samrocksnature, Nofancyname
Vasya, an invisible virus, is sitting on a square plate with a side of 
cm. He and the doctor Petya take turns. With the next 
-th move, Petya draws with the vaccine like ink a segment micron long, and then Vasya must choose a direction and crawl in this straight line a direction of 
micron distance (without going beyond the edge of the plate). If Vasya crawls through any of the points with the vaccine or touches it, he will die. Petya can act with any precision. Can Petya within a finite number of moves for sure kill the virus?
cm. He and the doctor Petya take turns. With the next 
-th move, Petya draws with the vaccine like ink a segment micron long, and then Vasya must choose a direction and crawl in this straight line a direction of 
micron distance (without going beyond the edge of the plate). If Vasya crawls through any of the points with the vaccine or touches it, he will die. Petya can act with any precision. Can Petya within a finite number of moves for sure kill the virus?This post has been edited 2 times. Last edited by parmenides51, Mar 7, 2021, 11:22 PM
chance that they are named Vasya and Petya.
th move. During his next 
aligned vertical segments at a distance of 
microns between any two consecutive ones, and another 
microns full of cells of size 
inside it, and all borders are drawn.
th move of the virus. If it is inside 
microns, and finally kills the virus.
,
microns. So he moves to inner part of a 
moves without success and eventually he's checking the board where Vasya the Virus is located. Then he draws 
.
microns away which exceeds 
,
. These moves are counted in the part
consecutive moves Petya can frame any unit square on the plane. He begins to frame the unit squares on the lattice points, so that he consecutively fills the squares with corners at 
for 
One move for every 
from the origin, where 
is a constant. At the same time, at its 
from the origin. Since the growth rate of 
is faster than 
the virus will be framed after finite amount of time, and later it will be killed. The only drawback is the doctor will not know exactly when this will happen.
