Thanks to Simon Foundation : a youtube video.
Here is one more problem for those of you love to think combinatorially 🙂
Consider a finite sequence of real numbers for which the sum of any seven consecutive terms is negative and the sum of any 11 consecutive terms is positive. Find the greatest number of terms of such a sequence.
Hint: Construct some concrete example and play with it!
You are welcome to send your solutions, comments, etc.
Prove that among any 10 entries of the table:
situated in different rows and different columns, at least two are equal.
Please send your comments, solutions…most welcome 🙂
Sequences of integers are a favourite of olympiad problem writers since such sequences involve several different mathematical concepts, including for example, algebraic techniques, recursive relations, divisibility and primality.
Consider the sequence defined by , and
for all . Prove that all the terms of the sequence are positive integers.
There is no magic or sure shot or short cut formula to such problems. All I say is the more you read, the more rich your imagination, the more you try to solve on your own.
Replacing n by yields, and we obtain
This is equivalent to
or for all . If n is even, we obtain
while if n is odd,
It follows that , if n is even,
and , if n is odd.
An inductive argument shows that all are positive integers.
Prove the following identity (a):
Prove the following identity (b):
Prove the following identity (c);
, if and is 0, otherwise.
General Hint for all three problems: Please see Wikipedia for Stirling numbers!
If x is an integer, both sides of the first identity count mappings of an n element set into an x-element set.
Both sides count pairs where is a permutation of an n-element set S, and is a coloration of S with x colors invariant under .
Combine the identities in (a) and (b).
Suppose first that x is a positive integer. Let and . The number of mappings of N into X is . On the other hand, let k denote the cardinality of the range of a mapping of N into X. For k fixed, we can specify in ways which elements of N are mapped onto the same element of X. Once this partition of N is specified, we have to find an image for each class of it, distinct images for distinct classes. This can be done is ways. Thus,
is the number of mappings of N into X with range of cardinality k. This proves the identity when x is a positive integer. But this means that if we consider x as a variable the polynomials on the two sides have infinitely many values in common. Therefore, they must be identical.
Again, we may assume that x is a positive integer. If a permutation of a set S has exactly k cycles, then
, then is the number of x-colorations of S invariant under . The left hand side of the identity sums these numbers for all permutations of . A given x-coloration of S is counted times, where is the number of elements with color i. The number of occurrences of a given sequence is
and so, (fill this missing gap in proof) this sum is
Homework and do send comments, suggestions, questions…
We have k distinct postcards and want to send them all to our friends ( a friend can get any number of postcards, including 0). How many ways can this be done? What happens if we want to send at least one card to each friend?
Decide now about the postcards. The answer to this question is .
I. We have to decide about the postcards independently. Any postcard can be sent to any of the n friends. Hence, the result if .
II. Let be the cards. The set must be split into n disjoint non-empty sets . Thus, is a partition of S. From any partition of S into n (non-empty) classes we get possibilities to send out the postcards. Hence, the answer is .