I found this results while doing my favourite activity - skimming through old books at library :) The first one is a result (n-above-a) and what follows is a theorem called Pasting Theorem in literature. n above a Let $a$ be a fixed real number. Then - Proof: $$\text{Part I}$$ Suppose first that $0 < a < 1.$ It follows that ${1 \over a} > 1$ and so we may write $${1 \over a} = 1 + h \quad \text{where } h > 0$$

I recently came across this question given in some olympiad paper and found it quite accessible, so here is the proof in all its glory after a lemma that will be in use. By the way, all the theorem says is that as long as $p \geqslant 1$ we can approximate any real number to a rational number by some arbitary precision. Lemma: Consider $n+1$ positive real numbers $x_0, x_1, x_2, x_3, \ldots, x_n$ all being in $[0,1)$ then for any $i \neq j$ and $i > j$ it so happens that $$\left | x_j - x_i \right | < \frac{1}{n}$$ for $i,j \in \mathbb{Z}^+$

In this post we take a dive into building a very minimalist toolkit which might be useful for Diophantine Equation problems. This write-up especially focusses on the easier side of things related to cater Scholarship problems and not Olympiad Problems which may require various additional specialised methods. We do not illustrate any methods with examples since it is often quite restricting to see a few examples and become limited to one such method, whereas in an exam environment multiple trial with several approaches are almost always necessary.

Often-a-while fibonacci sequence problems stars as the guest problem in olympiad exam papers and often the solution is hidden from the untrained eye as some form of application of its properties. This article assumes familiarity with the definition of the “Fibonacci Sequence” and the ability to comprehend recursive definitions at the minimum. We will present various properties related to Fibonacci Numbers (and related) without any proof. It is the interest of the reader to seek out the proofs and validate each result for their own.