be the incircle of triangle 
and 
at 
.
meets 
.
be 
-
be the midpoint of 
,
are concyclic.
and incentre 
.
intersects the lines 
,
,
at points 
,
,
,
,
,
,
of the segments 
,
,
.
and 
be prime numbers such that 
is perfect square. Prove that 
is prime
such that for all positive real numbers 
that satisfy the equation![\[
f(x) + 3f(-x) + f\left( \frac{1}{x} \right) = x, \quad \text{for all } x \in \mathbb{R}^*.
\]](http://latex.artofproblemsolving.com/6/a/2/6a2e35c42644036a62451b5e1794b36ffa728591.png)
is irreducible for every natural number 
.
and 
be concentric circles, with 
to 
)
and 
on 
.
where 
.
is the circumcenter of triangle 
is the projection of point 
onto line 
.
,
are 
,
,
,
intersects 
and 
at points 
and 
,
.
of positive odd integers, such that 
and 
.
,
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
Y by Rounak_iitr
Dear Mathlinkers,
1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.
Prove : UV is parallel to PM.
Sincerely
Jean-Louis
1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.
Prove : UV is parallel to PM.
Sincerely
Jean-Louis
is cyclic, and we also have 
cyclic, so 
.
