India National Olympiad 1986 Problems
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 exactly exchanged place, when did he go out ?
Problem 2
Solve
\[ \left\{ \begin{array}{l}
\log_2 x+\log_4 y+\log_4 z=2 \\
\log_3 y+\log_9 z+\log_9 x=2 \\
\log_4 z+\log_{16} x+\log_{16} y=2 \\
\end{array} \right.\]
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.
Problem 8
Suppose $ A_1,\dots, A_6$ are six sets each with four elements and $ B_1,\dots,B_n$ are $ n$ sets each with two elements, Let $ S = A_1 \cup A_2 \cup \cdots \cup A_6 = B_1 \cup \cdots \cup B_n$. Given that each elements of $ S$ belogs to exactly four of the $ A$’s and to exactly three of the $ B$’s, find $ n$.
Problem 9
Show that among all quadrilaterals of a given perimeter the square has the largest area.
