# A nice analysis question for RMO practice

Actually, this is a famous problem. But, I feel it is important to attempt on one’s own, proofs of famous questions within the scope of RMO and INMO mathematics. And, then compare one’s approach or whole proof with the one suggested by the author or teacher of RMO/INMO.

Problem:

How farthest from the edge of a table can a deck of playing cards be stably overhung if the cards are stacked on top of one another? And, how many of them will be overhanging completely away from the edge of the table?

Reference:

I will post it when I publish the solution lest it might affect your attempt at solving this enticing mathematics question !

Please do not try and get the solution from the internet.

Regards,

Nalin Pithwa.

# A miscellaneous algebra question for RMO or Pre-RMO

Question:

If a, b, c are three rational numbers, then prove that

$\frac{1}{(a-b)^{2}} + \frac{1}{(b-c)^{2}} + \frac{1}{(c-a)^{2}}$

is always the square of a rational number.

Solution to be posted soon…

Cheers,

Nalin Pithwa.

# Euler’s Series: problem worth trying for RMO or INMO or Madhava Mathematics Competition

Mengoli posed the following series to be evaluated:

$1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}}+\ldots$

Some great mathematicians, including Leibnitz, John Bernoulli and D’Alembert, failed to compute this infinite series. Euler established himself as the best mathematician of Europe (in fact, one of the greatest mathematicians in history) by evaluating this series initially, by a not-so-rigorous method. Later on, he gave alternative and more rigorous ways of getting the same result.

Question:

Can you show that the series converges and gets an upper limit? Then, try to evaluate the series.

Solution will be posted soon.

Cheers,

Nalin Pithwa.

# Inequalities and mathematical induction: RMO Homi Bhabha sample questions:

Prove by mathematical induction, the following:

1) $2^{n}(n!)^{2} \leq (2n)!$ for all $n \geq 1$.

2) Establish the Bernoulli inequality: If $(1+a)>0$, then $(1+a)^{n} \leq 1+na$ for all natural numbers greater than or equal to 1.

3) For all $n \geq 1$ with $n \in N$ prove the following by mathematical induction:

a) $\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \ldots + \frac{1}{n^{2}} \leq 2-\frac{1}{n}$

Solutions will be put up tomorrow!

Nalin Pithwa.