AoPS Academy
. Get all the 
such that 
is uniformly continuous on 
.
satisfied
for all 
and all 
.
be defined by 
(a) Show that 
has exactly one point of discontinuity. at its point of discontinuity.
times differentiable real function 
has at least 
distinct zeros in the closed interval ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
and that the polynomial 
has only real zeroes. Show that
has at least one zero in , where 
denotes the -th derivative of 
be a closed map with the property that the inverse image of each point in 
is a compact subset of 
. Prove that 
is compact whenever 
is compact in .
denote the sum of the alternating harmonic series. Rearrange this series as follows
Assume as known that this series converges as well and denote its sum by 
. Denote by 
respectively the 
-th partial sums of both series. Prove that
has always integral solutions for 
and 
whenever 
is a positive integer.
is a sequence of real numbers such that
Prove that if 
converges, then 
for all .
where 
, is cut by the line 
the tangents at the points of intersection are concurrent.
