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.
an increasing sequence, with 
, if 
then 
increasing, from a certain rank.
?
the points 
and 
are the feet of the altitudes through 
and 
respectively. The incenters of the triangles 
and 
are 
and 
respectively; the circumcenters of the triangles 
and 
are 
and 
respectively. Prove that 
and 
are parallel.
with 
prime and ![\[a^p+b^p=p!.\]](http://latex.artofproblemsolving.com/f/e/b/feb27894b1f6497559eeb094cfe92256bc26688e.png)
such that 
. Show that 
. When will equality hold?
be a cyclic quadrilateral with circumcenter 
, such that 
is not a diameter of its circumcircle. The lines 
and 
intersect at point 
, so that 
lies between 
and , and 
lies between and . Suppose triangle 
is acute and let 
be its orthocenter. The points 
and on the lines and , respectively, are such that 
and 
. The line through , perpendicular to , intersects at 
, and the line through , perpendicular to , intersects at 
. Prove that the points , , are collinear.
which satisfy the condition![\[ f(m + n) \geq f(m) + f(f(n)) - 1
\]](http://latex.artofproblemsolving.com/4/b/d/4bddce124cf0da82cb8bb6c6ae2325239354f27c.png)
Find all possible values of 
be a positive integer and 
be a positive integer coprime to 
. Let 
, and for 
, define
Find, in terms of and 
, the greatest positive integer 
for which there exists an index 
such that 
is divisible by 
.
table are initially white. Alice and Bob play a game. First Alice paints of the fields in red. Then Bob chooses 
rows and columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of such that Alice can win the game no matter how Bob plays.
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is a continuously differentiable function such that 
and 
for some 
. Prove that there exist two tangents to the graph of 
that form an equilateral triangle with an appropriate segment of the 
-axis.
such that 
(otherwise would be strictly increasing or decreasing over the whole interval, contradicting 
). Hence there exists 
for which 
, because the derivative has the intermediate value property. Now the absolute maximum of must occur at the location of such an , hence there actually exists an 
for which and 
. Then if 
denotes the slope function, note that 
and by the mean value, intermediate value properties there exists 
such that 
. The corresponding tangents to the graph of running through 
and 
(i) intersect the x-axis at 
angles, (ii) do not intersect on the x-axis, and (iii) are not parallel hence intersect (somewhere off the x-axis). This is the desired equilateral triangle then.
is continuous. And given By Rolle's theorem there exist 
such that 
. Now there is a such that 
. So 
. Now applying LMVT on 
we have a 
such that
Similarly applying LMVT on 
we have a 
such that
Now apply IVT on 
we have a 
such that 
and similarly we have a 
such that 
. Easy to see that tangents at 
makes an equilateral triangle with finite segment on x axis.
such that 
.
