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 a sequence of positive real numbers that satisfies 
and 
for every natural number 
. Prove that 
for every natural number .
of real numbers such that![\[ a_{m^2 + m + n} = a_{m}^2 + a_m + a_n\]](http://latex.artofproblemsolving.com/b/b/9/bb97053e86e7733fe5c26f7859c87566eb444b52.png)
for all positive integers 
and .
be an acute triangle with 
, midpoint 
of side 
, altitude 
(
), and orthocenter 
. A circle passes through points 
and 
, is tangent to line 
, and intersects the circumcircle of triangle at a second point 
. The circumcircle of triangle 
intersects line 
at a second point 
. Prove that the lines 
and 
are perpendicular.
such that for all positive real numbers 
,![\[x^2[f(x)+f(y)]=(x+y)f(yf(x)).\]](http://latex.artofproblemsolving.com/2/7/c/27c95af9b083a2a2827d428da24a5152fd926473.png)
Determine all sets 
containing 
and having the property that 
is prime, for all distinct 
be nonnegative real numbers such that 
. Find min
such that the number 
is a perfect square for all positive integers .
and 
be two similar right-angled triangles (in the same plane) with 
such that 
and 
lie on opposite sides of the line 
. If 
, calculate the ratio 
.
. Prove that:
2/ Let . Prove that: 
![$f:[0,1]\rightarrow [0,1].$](http://latex.artofproblemsolving.com/d/0/7/d076b5242cc57c3c4af7971c9cb104023effca65.png)
I'm interested in what solutions there are other than constant functions.
the angle at vertex 
measures 
. Point 
is the foot of the perpendicular from point 
to the line 
. If 
and
, find the area of 
.
)
and 
Prove that
?
and 
.![\[ab+bc+ac\leqslant 27\]](http://latex.artofproblemsolving.com/c/b/5/cb5e2a1efd462abd2354f4136f5fa64408f3138c.png)
. Prove that
