AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
such that ![\[f(x+f(x)+2f(y))+f(2f(x)-y)=4x+f(y)\]](http://latex.artofproblemsolving.com/4/4/2/4423651bb829b92debd65a23f6cee3d4d4b8530f.png)
holds for all reals 
and 
.
, such that: 
For all 
.
be positive real numbers such that : 
. Prove that : 
be an odd integer. Prove that there exists a prime number 
such that![\[
p \mid 2^{\varphi(n)} - 1 \quad \text{but} \quad p \nmid n.
\]](http://latex.artofproblemsolving.com/3/5/1/3514b64ecd5f7b2586d5bfeae59b7c9504d5b455.png)
be a triangle for which there exists an interior point 
such that 
. Let the lines 
and 
meet the sides 
and 
at 
and 
respectively. Prove that ![\[ AB+AC\geq4DE. \]](http://latex.artofproblemsolving.com/3/a/4/3a49c5f8bd2930adca7bde2601dd045d4bf05ff6.png)
such that ![\[f(xf(y)+f(x+y)+1)=(y+1)f(x+1)\]](http://latex.artofproblemsolving.com/4/9/e/49ec422b640766de653d9d640bbc45065a6ac375.png)
holds for all 
.
on one variable with real coefficients such that the equation 
has exactly 
ordered pairs 
as solutions?
such that
for all nonnegative real numbers 
, 
, and 
with 
.
be three non-negative real numbers satisfying 
such that any real numbers and satisfy
Proposed by EmilXM
board has some of its cells coloured red, blue, or green. Each cell is coloured with at most one colour, and some cells may remain uncoloured. Additionally, there is at least one cell of each colour. Two coloured cells are said to be friends if they have different colours and lie in the same row or in the same column. The following conditions are satisfied:
now use ceva's therem and use the fact that if 
then 
.
,
,
,
,
.
.
such that 
of the triangle 
intersects the side 
and 
in 
,
.
and 
.![\[ a(1+p)\cdot \overrightarrow{IP}=(a-pb)\overrightarrow{IB}-pc\overrightarrow{IC}\]](http://latex.artofproblemsolving.com/c/2/6/c2683ffb904348a083448c90f588aeabae66b513.png)
.
,
and 
are concurrents.
