INMO 1986

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 ?

Solve $$\left\{ \begin{array}{l} \log_2 x+\log_4 y+\log_4 z=2 \log_3 y+\log_9 z+\log_9 x=2 \log_4 z+\log_{16} x+\log_{16} y=2 \end{array} \right.$$

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

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.

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

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