India National Olympiad 1987 Problems
Problem 1
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.
Problem 2
Determine the largest number in the infinite sequence
\[ 1, \sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4}, \dots, \sqrt[n]{n},\dots\]
Problem 3
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.
Problem 4
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.
Problem 5
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$.
Problem 6
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.
Problem 7
Construct the $ \triangle ABC$, given $ h_a$, $ h_b$ (the altitudes from $ A$ and $ B$) and $ m_a$, the median from the vertex $ A$.
Problem 8
Three congruent circles have a common point $ O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the common point $ O$ are collinear.
Problem 9
Prove that any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite side) is isosceles.
