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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.
be an odd integer greater than 
Let 
be an 
symmetric matrix such that each row and column consists of some permutation of the integers 
Show that each of the integers 
must appear in the main diagonal of .
convergent?
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
?