INMO 1987

Given $m$ and $n$ as relatively prime positive integers greater than one, show that $$\frac{\log_{10} m}{\log_{10} n}$$ is not a rational number.

Determine the largest number in the infinite sequence $$1, \sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4}, \dots, \sqrt[n]{n},\dots$$

Let $T$ be the set of all triplets $(a,b,c)$ of integers such that $1 \leq a < b < c \leq 6$ For each triplet $(a,b,c)$ in $T$ , take number $a\cdot b \cdot c$ . Add all these numbers corresponding to all the triplets in $T$ . Prove that the answer is divisible by 7.

If $x$ , $y$ , $z$ , and $n$ are natural numbers, and $n\geq z$ then prove that the relation $x^n + y^n = z^n$ does not hold.

Find a finite sequence of 16 numbers such that: (a) it reads same from left to right as from right to left. (b) the sum of any 7 consecutive terms is $-1$ , (c) the sum of any 11 consecutive terms is $+1$ .

Prove that if coefficients of the quadratic equation $ax^2+bx+c=0$ are odd integers, then the roots of the equation cannot be rational numbers.