[/quote]
,
which satisfy the following set of equations
inscribed in 
with 
being the midpoint of 
.
of 
.
be the projection of 
onto 
.
that is not on 
intersects 
at 
.
and 
intersect at 
.
is an isosceles triangle.
,
be positive real numbers, such that 
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
such that 
divides 
for any positive integers 
and 
.
of real numbers such that 
for all positive integers 
.
denotes the greatest integer less than or equal to 
.
distinct real numbers there exist two pairs 
and 
with 
or 
,![\[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]](http://latex.artofproblemsolving.com/3/b/0/3b03250ffbd03fd657fa00bc2b63a2c17ee75061.png)
be an arbitrary diameter of its circumscribed circle, and the point 
lies on the smaller arc 
,
is on the smaller arc 
.
are feet of perpendiculars dropped from point 
to line 
lies on a fixed circle.
such that
,
be an abitary prime number then we can find out that if 
then 
then they say that exist integer 
such that 
i don't know why 
,
such that![\[ x f(x + xy) = x f(x) + f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.\]](http://latex.artofproblemsolving.com/b/0/f/b0f1a1d3b9de5d23f1b5a2e9be3b88d8bd575883.png)
be points on the sides 
respectively of a triangle 
and 
meet at 
and 
,
and 
,
,
.
to the area of 
.
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
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 
.
