1986 INMO

This was the first INMO conducted, it was conducted by the Indian Institute of Science, Department of Applied Mathematics along with the National Board of Higher Mathematics, Department of Atomic Energy(NBHM DAE) .It had 9 problems to be completed in 3 hours.

Problem 1

A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exchanged places. When did he go out ?

Problem 2

Solve:

\begin{cases} \log_{2} x + \log_{4} y + \log_{4} z = 2 \\[6pt] \log_{3} y + \log_{9} z + \log_{9} x = 2 \\[6pt] \log_{4} z + \log_{16} x + \log_{16} y = 2 \end{cases}

Problem 3

Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles Prove that:

$\frac{1}{\sqrt{c}} = \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$

Problem 4

Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number

Problem 5

If P(x) is a polynomial with integer coefficients and a, b, c three distinct integers, then show that it is impossible to have P(a)=b, P(b)=c, P(c)=a.

Problem 6

Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal

Problem 7

If a,b,x,y are integers greater than 1, such that a and b have no common factor except 1 and $x^a = y^b$ , show that $x = n^b, \; y = n^a$ for some integer n greater than 1