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be a scalene triangle with orthocenter 
and circumcenter 
. Denote by 
, 
the midpoints of 
, 
. Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
.
Prove that
Where 
Find themaximum value of 
Find themaximum value of 
is 
.
and 
for 
, both conditions are met with the desired range. We now show that this is the smallest possible attainable value. By the conditions there must exist indices 
and 
such that 
. One of 
must be at least 
so the the maximum of the sequence is at least . On the other hand, 
Hence one of these terms must be less than 
. Thus the maximum minus the minimum in this case is at least as wanted.
. Then we have 
So one of these terms is at least 
. Also since the sum of the first terms is 
, one of them must be at most , finishing this part of the problem.
is 
.
, 
, and 
for 
, both conditions are met with the desired range. We now show that this is the largest possible attainable value. Begin by noticing that 
for all 
. Suppose 
is the maximum and 
is the minimum. Then 
finishing the problem.
Prove that
Where 
.
be a positive integer. Let 
be a sequence of real numbers such that![\[\max_{1 \leq k \leq n} \frac{x_1 + \dots + x_k}{k} = 1 \quad \text{and} \quad \min_{1 \leq k \leq n} \frac{x_1 + \dots + x_k}{k} = 0.\]](http://latex.artofproblemsolving.com/f/1/a/f1afd7d53391f845f9cb0fdedb5b814527164cc8.png)
Find the minimum and maximum possible values of 
.
