Let $$ABC$$ and $$PQR$$ be any two triangles in the same plane. Assume that the prependiculars from the points $$A, B, C$$ to the sides $$QR, RP, PQ$$ respectively are concurrent. Using vector methods or otherwise, prove that the prependiculars from $$P, Q, R $$ to $$BC,$$ $$CA$$, $$AB$$ respectively are also concurrent.

For points $$P\,\,\, = \left( {{x_1},\,{y_1}} \right)$$ and $$Q\,\,\, = \left( {{x_2},\,{y_2}} \right)$$ of the co-ordinate plane, a new distance $$d\left( {P,\,Q} \right)$$ is defined by $$d\left( {P,\,Q} \right)$$$$ = \left( {{x_2},\,{y_2}} \right)\left| {{x_1} - {x_2}} \right| + \left| {{y_1} - {y_2}} \right|.$$ Let $$O = (0, 0)$$ and $$A = (3, 2)$$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $$O$$ and $$A$$ consists of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.

The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

A coin probability $$p$$ of showing head when tossed. It is tossed $$n$$ times. Let $${p_n}$$ denote the probability that no two (or more) consecutive heads occur. Prove that $${p_1} = 1,\,\,{p_2} = 1 - {p^2}$$ and $${p_n} = \left( {1 - p} \right).\,\,{p_{n - 1}} + p\left( {1 - p} \right){p_{n - 2}}$$ for all $$n \ge 3.$$

Prove by induction on, that $${p_n} = A{\alpha ^n} + B{\beta ^n}$$ for all $$n \ge 1,$$ where $$\alpha $$ and $$\beta $$ are the roots of quadratic equation $${x^2} - \left( {1 - p} \right)x - p\left( {1 - p} \right) = 0$$ and $$A = {{{p^2} + \beta - 1} \over {\alpha \beta - {\alpha ^2}}},B = {{{p^2} + \alpha - 1} \over {\alpha \beta - {\beta ^2}}}.$$