[/quote]
,
be a sequence of positive numbers satisfying, for any positive integers 
such that 
,
Show that there exist positive numbers 
so that 
for any positive integer 
.
and circumcentre 
.
be the touchpoints of the incircle with 
,
is the Euler line of 
.
and medial triangle 
.
be a point on 
.
is the midpoint of segment 
,
is isoceles.
such that the sequence 
defined by ![\[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \]](http://latex.artofproblemsolving.com/2/1/b/21b2a1f93b11a94b84e7b55f4b4f679aa20e36c6.png)
contains at least one integer term.
non-negative real numbers,but 
0![\[1\leq \frac{a+b}{a+4b+c}+\frac{b+c}{b+4c+a}+\frac{c+a}{c+4a+b}\leq \frac{4}{3}\]](http://latex.artofproblemsolving.com/8/b/7/8b7d7b8cbcab8f2eed31e17aac717fa3c70c92f1.png)
be the point such that the quadrilateral 
is a parallelogram. Let 
such that 
.
at 
and 
.
.
that satisfy the following property: if 
,
,
,
are all the positive divisors of 
,
divides 
for every 
.
.
,
and 
.
is tangent to 
.
be the midpoints of segments 
intersects 
and 
,
.
,
is the point on 
.
at 
be the second intersection of 
.
be the circumcircle of 
be the midpoint of the arc 
of 
at 
.
at 
by 
.
.
.
be natural numbers. As the New Year arrives, it is a cherished tradition among Iranian families to call their relatives and exchange Nowruz greetings. Just moments after the New Year begins, a family with 
members to celebrate the occasion. Since the two families are close, every member of one family wishes to speak with every member of the other family to exchange greetings. Thus, during this single call, all 
distinct one-on-one conversations must take place exactly once.
.
be its circumcircle, 
such that 
and let 
.
and 
are tangent to each other.
Y by
Let 
be an acute-angled triangle. Let 
be the midpoint of the arc 
which contain the point 
. Let 
and 
be the foot(s) of the perpendicular(s) of the point to the lines 
and 
, respectively. Define 
analogously.
a) Prove that the line
cuts in the midpoint.
b) Prove that the lines
and 
are concurrents.
be an acute-angled triangle. Let 
be the midpoint of the arc 
which contain the point 
. Let 
and 
be the foot(s) of the perpendicular(s) of the point to the lines 
and 
, respectively. Define 
analogously.a) Prove that the line
cuts in the midpoint.b) Prove that the lines
and 
are concurrents.This post has been edited 1 time. Last edited by mathisreal, Feb 14, 2022, 9:52 PM
the midpoints of 
.
is perpendicular to 
,
are in the simson line of the point 
is a simson line and 
is a simson line.
,
,
is parallel to the angle bisector of 
.
.
is the simson line of the triangle ABC relative to verex A. So, It suffices to show that the projection of 
,
, C = 
and 
.
but this simplifies to 
,
.
is parallel to the internal bissector going trought A. This is proven checking if the angular coefficients of the lines are equal. The angular coefficient of the line ab is 
- 
Superior part of the fractions
)
Inferior part of the fraction
and do some simplifications.
superior part of the fraction
inferior part of the fraction
. And this let us left with 
.
that is 
are homothetic to the triangle ABC and their respective lines are parallel, the Simson lines meet themselves at the incenter of 
