Volume 27


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

In experimental science in past eras data was collected from an experiment and some relationship between quantities in the form of equations was sought.

Q.852 If $a_1,a_2,\cdots a_n$ are positive real numbers and $a_1+a_2+\cdots + a_n =1$ prove that $$ \sum_{k=1}^n \left(a_k + \frac{1}{a_k}\right)^2 \leq \frac{(n^2+1)^2}{n}.$$

Q.840 (i) Let $\alpha ,\beta$ be two distinct solutions of $$x^3 - x^2 - x + c = 0.$$ Simplify $\alpha^2\beta + \alpha\beta^2 - \alpha\beta$.