[/quote]
,
such that 
divides 
for all positive integers 
and 
with 
.
,
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
be an acute triangle with 
,
be the intouch triangle with 
,
,
,
be the intersecttion of the perpendicular from 
to 
with 
,
.
and 
are concylic
,
,
.
and that 
.
and 
be the midpoints of sides 
,
and 
be the feet of the altitudes from 
and 
in 
,
be the intersection of line 
with the tangent line to 
.
be the intersection point, other than 
.
be the intersection of lines 
and 
.
with centre 
passes through 
respectively. Let 
,
is equidistant from 
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
,
,
,
,
satisfy 
and 
.
of 
such that 
.
and 
)
Prove that if lines 
are similar.
such that 
, where 
are non-negative reals such that 
.
.
be lengths of tangents from 
to the incircle of the quadrilateral which center is 
.
be the midpoints of 
resp. Prove that![\[ \frac{MI}{IN}=\frac{a+c}{b+d} \]](http://latex.artofproblemsolving.com/e/7/0/e709acb8b17c0466a385ba8e7a89bb1bf16c72af.png)
is given. It turns out that for every positive integer number 
,
and 
are positive, they have the same number of divisors. Is it necessarily true that 
?
Y by
Is it true that
?
A=1728400904217815186787639216753921417860004366580219212750904
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336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975
B=2275643401548081847207782760491442295266487354750527085289354
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953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
?A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975
B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
This post has been edited 6 times. Last edited by Dattier, Yesterday at 10:10 AM
.
