High School Mathematics Extensions/Mathematical Proofs/Problem Set/Solutions

Mathematical Proofs Problem Set 1. 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: 3. 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. Alternate solution Notice that letting a = b = 1, we get as required. 5. 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.