Volume 7
, Issue 2

1971

Early in the 17th century, Johannes Kepler established, from actual observations of the positions of the planets in the sky, three laws of planetary motion.

A Mersenne number is an integer of the form $2^p-1$ where $p$ is a prime.

$x$ and $y$ are unequal positive integers. Prove that $xy$ does not divide x^2 +y^2. \begin{align*} 1+\cos(x) & = 1 + (1-\sin^2(x))^{1/2} \\ (1+\cos(x))^2 & = \left\{1 + (1-\sin^2(x))^\frac{1}{2}\right\}^2 \\ \end{align*} Letx=\pi$. "Murgatroyd's Mind-stretchers" by J. and F. Pinkney Suppose you have two glasses; one contains water and the other contains the same amount of cordial. J151 (i) Prove that if$k$is not a prime then neither is$2^k-1$J141 The real numbers$a,b$and$c\$ are such that $$a^2 + 4b^2 + 9c^2 = 2ab + 6bc + 3ca.$$ Prove that $$a = 2b = 3c.$$