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such that 
is a square.
such that there is a polynomial 
with integer coefficients for which 
and where 
is a prime number .
letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
defined on the positive reals such that 
for all 
.
such that 
is a perfect square, i.e., the square of an integer.
, without repetition. Find the largest multiple of 
. Justify your answer.
such that 
, for all 
.
satisfying the system of equations:
, compute the ratio 
. Prove that sum of these ratios is at least 
.
is given by 
.
without satisfying 
.
is inscribed a cylinder with base lying on the base plane of the half-ball, and another such cylinder with equal volume. If the base-radius of the first cylinder is 
, what is the base-radius of the other one?
is inscribed a cylinder with base lying on the base plane of the half-ball, and another such cylinder with equal volume. If the base-radius of the first cylinder is , what is the base-radius of the other one?
