Problem 1) Prove the identity

What do we get if ?

*Hint: Count the number of mappings of a set of k given elements onto a set of n given elements.*

Problem 2) Prove the following Abel identities:

Problem 2a)

Problem 2b)

Problem 2c)

*Hint: To prove problem 2a, differentiate both sides with respect to y, and use induction on n. The other two sub-problems will follow from this sub-problem a.*

Problem 3) Let

Prove that .

Find other examples of such sequences of polynomials.

*Hint: Use *

Problem 4) Evaluate the determinants:

and

where denotes the greatest common divisor of i and j.

Problem 5) The number of non-congruent triangles with circumference , and integer sides is equal to the number of non-congruent triangles with circumference and integer sides. This number is also equal to the number of partitions of n into exactly three terms. Determine the number.

More conundrums to follow…(no hints for problems 4 and 5 above ! :-))

*Nalin Pithwa*

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