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
.
denote the number of positive integers less than 
that are relatively prime to . Prove that there exists a positive integer 
for which the equation 
has at least 
solutions in .
is a rational number.
and other permutations.
with real coefficients for which there exists a polynomial 
with real coefficients such that for all real 
, 
of points in the plane is balanced if, for any two different points 
and 
in , there is a point 
in such that 
. We say that is centre-free if for any three different points , and in , there is no points in such that 
.
, there exists a balanced set consisting of points. for which there exists a balanced centre-free set consisting of points.
, such that every positive integer appears exactly once as a member of the sequence and 
divides 
if and only if 
?
be a sequence of real numbers, and let be a fixed positive integer less than . We say an index 
with 
is good if there exists some 
with 
such that 
, where the indices are taken modulo . Let 
be the set of all good indices. Prove that 
.
such that for any positive real numbers and 
,
be a twice differentiable function such that![\[
f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1.
\]](http://latex.artofproblemsolving.com/1/e/1/1e178debf8ae0b6b76d2e60ffca95f5a4c4ad4e5.png)
Prove that there exists a point 
such that![\[
f(\xi) \cdot f'(\xi) + f''(\xi) = 0.
\]](http://latex.artofproblemsolving.com/4/6/b/46b8d103981b838ae0aa5afdee104edfae4e4b8f.png)
be a function such that 
and for 
Prove that
exists and is less than 
be a function, two times derivable on 
for which there exist 
such that![\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\]](http://latex.artofproblemsolving.com/a/3/1/a31861ef909a6e17ed8b495b9b9b16d887ba573f.png)
for all 
.
.
defined for 
being an interval (connected subset of the real line) and 
not being in this image, it follows that we may assume that 
for all . But then 
for all 
and so 
is a minimum point of 
. Of course, this can be written in 11-th grade vocabulary. 
has the same sign for all 
.Can you be give me more detailes please 
?The official solution is based on the fact that the function 
is injective ,hence monotonic.
such that is a connected subset of the plane and the function 
is continuous on this domain, thus its image is a connected subset of the line, thus an interval. I agree however that this is not a solution of an 11-th grade student, but that's how life is. I will not be amazed if in a few years I see complex analysis, Lebesgue integration and other such stuff at RMO. It's quite à pity, it gives huges advantages to some people. 
should be defined on the set of pairs such that 
to ensure that its domain is connected. Of course, all the rest works with this modification, I don't know how I could write such a stupid thing. 
, then 
is strictly concave up or down on some neighbourhood of the point , thus one can draw a close enough parallel to the line tangent at to the function's graph that intersects the graph in two points 
and 
. The slope of that tangent would be 
, equal to the slope of the tangent at , that is,.
is connected, but why is connected ? has clearly two connected components. However with this spirit we can also solve the problem, extend 
to whole plane defining 
and now 
is continuous on whole plane and do similar thing.
has the same sign for all .Can you be give me more detailes please ?The official solution is based on the fact that the function is injective ,hence monotonic.
there exists an with the desired property. So you get 
only for values in some set 
, which has to be dense in the reals, but need not be an interval, so the IVP you use later on in your proof does not give a contradiction. (See the solutions above for proofs that avoid this problem.)
is continuous on 
and that is true because 
is twice differentiable.
. In the first paragraph you show that for all 
has to contain some 
(which means that is a dense subset of the real numbers). This part of your proof is fine. But in order to make the proof of the green claim work you need to show that contains all real numbers. This is missing in your proof (if I interpret you proof correctly).
.
,
for all 
and now make 
close to 