Monday, May 26, 2014

Puzzle 49

A rectangular sheet of paper ABCD is with width AD=1 and length AB more than the width but not more than double of it. The paper is folded through vertex A so that the edge along AD falls onto AB. Without unfolding, the paper is folded again along a line through B so that CB now lies on AB. Now the result is a triangular piece of paper. There is a region on this triangular sheet that has four layers of paper. Find the area of that region in terms of the length x, of the rectangle.

Saturday, May 24, 2014

Puzzle 48 - Roots of a cubic

From Rolle’s theorem, we can conclude that between every pair of distinct roots of a polynomial P(x), there lies a root of its derivative P’(x). If P’(x) keeps its sign unchanged between any two distinct real numbers, then it is possible that P(x) does not have a root between them and hence may keep the same sign between those real numbers. Further if P(x) has two equal roots, then it will have a common root with its derivative. Consider with being an integer between and both inclusive, then find the number of values of if has to have
(i) one real root and two complex roots, (ii) only two equal roots