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.
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:
be an acute-angled triangle with 
and let 
be the centre of its circumcircle 
. Let be a point on the line segment 
such that 
. Let be the second point of intersection of and the line 
. If 
, 
and are the midpoints of the line segments 
, 
and , respectively, show that the points , and are collinear.
be two positive real numbers such that 
. Prove that
such that for any ,![\[f(xf(y)) + xf(x - y) = x^2 + f(2x).\]](http://latex.artofproblemsolving.com/1/d/4/1d47636bc583ed4732cf2ea5723bc8e49c70b80b.png)
Proposed by hyay
be the sequence of all natural numbers which are sum of squares of two natural numbers.
which 
.
a scalene triangle and 
your angle bisectors, with in the segment 
in the segment and in the segment . If 
.
.
and 
.
is cyclic 
is cyclic.
is the incenter of 
, then 
are collinear, implying that, by Desargues theorem on triangles 
and 
that 
are concurrent on a point 
on the radical axis of 
and 
, implying that 
is cyclic by power of a point. Thus, since AHCG is cyclic,
.
.
such that 
