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(DAE) .It had 9 problems to be completed in 3 hours.
Contents
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:
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 , show that
for some integer n greater than 1
Problem 8
Suppose are six sets, each with four elements, and
are
sets, each with two elements.
Let
It is given that each element of belongs to exactly four of the
’s and to exactly three of the
’s. Find
.
Problem 9
Show that among all quadrilateral of a given Perimeter the square has the largest area