| A1.
The roots of the equation x3 - x + 1 =
0 are a, b, c. Find a8 + b8
+ c8. |
| A2.
Find all real x which satisfy (x3 +
a3)/(x + a)3 +
(x3 + b3)/(x +
b)3 + (x3 +
c3)/(x + c)3 + 3(x - a)(x -
b)(x - c)/( 2(x + a)(x + b)(x + c) ) = 3/2.
|
| A3.
ABCD is a tetrahedron. The three edges at B are
mutually perpendicular. O is the midpoint of AB
and K is the foot of the perpendicular from O to
CD. Show that vol KOAC/vol KOBD = AC/BD iff
2·AC·BD = AB2. |
| B1.
Find all terms of the arithmetic progression -1,
18, 37, 56, ... whose only digit is 5. |
| B2.
Show that the sum of the maximum and minimum
values of the function tan(3x)/tan3x on
the interval (0, π/2) is rational. |
| B3. L
is a fixed line and A a fixed point not on L. L'
is a variable line (in space) through A. Let M be
the point on L and N the point on L' such that MN
is perpendicular to L and L'. Find the locus of M
and the locus of the midpoint of MN.
|