[/quote]
,
be positive real numbers such that 
.
.
and 
Prove that
be a scalene triangle with circumcenter 
,
,
.
,
,
at points 
,
,
respectively. Let 
be the projection of 
.
intersects the circumcircle of 
.
and 
intersect 
and 
respectively. Prove that the lines 
,
,
are concurrent.
be positive real numbers such that : 
. Prove that : 
and 
such that 
.
,
,
satisfying 
.
be an acute triangle with 
.
be the centroid of triangle 
and let 
to side 
.
intersects the circumcircle 
of triangle 
.
be the point where 
intersects the circle 
,
and 
be non-negative real numbers such that 
Prove that
and circumcenter 
and 
is a parallelogram. Prove that 
.
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](http://latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
![\[
(a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4.
\]](http://latex.artofproblemsolving.com/2/0/9/209f9807481e7a1e97f023bf65a603cb462c62cb.png)
(Karl Czakler)
that can be written in the form![\[
n = \frac{a^2 - b^2}{b},
\]](http://latex.artofproblemsolving.com/f/9/0/f90a3ce2e1b51d6e1cb53a15cad6f36c2b30492a.png)
where 
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
can be erased as 
,
as 
,
as 
.
s
n
t
hats, numbered from 
hats. After that, each wise man (in turn: first the first, then the second, etc.) loudly calls a number, trying to guess the number of his hat. The numbers mentioned should not be repeated. When all the wise men have spoken, they take off their hats and check which one of them has guessed. Can the sages to act in such a way that at least three of them knowingly guessed?
Y by Adventure10, Mango247
Dear Mathlinkers,
1. ABC a triangle
2. D the foot of the A-inner bissector
3. E the midpointAC
4. P the point of intersection of BE and AD
5. F the point of intersection of CP and AB
6. Pb the parallel to CF through B
7. M the point of intersection of Pb and DF
8. X the symmetric of B wrt AD.
Prove : MX is parallel to AB.
Sincerely
Jean-Louis
1. ABC a triangle
2. D the foot of the A-inner bissector
3. E the midpointAC
4. P the point of intersection of BE and AD
5. F the point of intersection of CP and AB
6. Pb the parallel to CF through B
7. M the point of intersection of Pb and DF
8. X the symmetric of B wrt AD.
Prove : MX is parallel to AB.
Sincerely
Jean-Louis
This post has been edited 1 time. Last edited by jayme, Aug 1, 2018, 11:34 AM
Reason: typo
Reason: typo
.
,
is a parallelogramm and we are done.
.
hence 
.
is a parallelogramm, and thus 
,
