High School Mathematics Extensions/Mathematical Proofs/Problem Set/Solutions
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Mathematical Proofs Problem Set 
- For all
- Therefore , , ...
- When a>b and c>d, a+c>b+d ( See also Replace it if you find a better one).
- Therefore we have:
- Let us call the proposition
- be P(n)
- Assume this is true for some n, then
- Now using the identities of this function:(Note:If anyone find wikibooks ever mentioned this,include a link here!),we have:
- Since for all n,
- Therefore P(n) implies P(n+1), and by simple substitution P(0) is true.
- Therefore by the principal of mathematical induction, P(n) is true for all n.
letting a = b = 1, we get
- Let be a polynomial with x as the variable, y and n as constants.
- Therefore by factor theorem(link here please), (x-(-y))=(x+y) is a factor of P(x).
- Since the other factor, which is also a polynomial, has integer value for all integer x,y and n (I've skipped the part about making sure all coeifficients are of integer value for this moment), it's now obvious that
- is an integer for all integer value of x,y and n when n is odd.