1962 IMO Problems

Day I

Find the smallest natural number $n$ which has the following properties:

(a) Its decimal representation has 6 as the last digit.

(b) If the last digit 6 is erased and placed in front of the remaining

digits, the resulting number is four times as large as the original number $n$ .

Determine all real numbers $x$ which satisfy the inequality:

$\sqrt{\sqrt{3-x}-\sqrt{x+1}}>\dfrac{1}{2}$

Consider the cube $ABCDA'B'C'D'$ ( $ABCD$ and $A'B'C'D'$ are the upper and

lower bases, respectively, and edges $AA'$ , $BB'$ , $CC'$ , $DD'$ are

parallel). The point $X$ moves at constant speed along the perimeter of the

square $ABCD$ in the direction $ABCDA$ , and the point $Y$ moves at the same

rate along the perimeter of the square $B'C'CB$ in the direction

$B'C'CBB'$ . Points $X$ and $Y$ begin their motion at the same instant from

the starting positions $A$ and $B'$ , respectively. Determine and draw the

locus of the midpoints of the segments $XY$ .