MathDB

1

sin x +1/2 sin 2x +1/3 sin 3x > 0

x

of the interval

0 < x <\pi

the inequality

\sin x +\frac12 \sin 2x +\frac13 \sin 3x > 0

holds.

4

1

digits of 16 naturals form a geometric sequence

Sixteen natural numbers written in the decimal system may form a geometric sequence, of which the first five members have nine digits, five further members have ten digits, four members have eleven digits and two terms have twelve digits. Prove that there is exactly one sequence with these properties.

3

1

a f(x^n) + f(-x^n) = b

Specify all functions

y = f(x)

, each in the largest possible domain (within the range of real numbers) of the equation

a \cdot f(x^n) + f(-x^n) = bx

suffice, where

b

is any real number,

n

is any odd natural number and

a

is a real number with

|a| \ne 1

.

1

4x4 system, x_1 + ax_2 + x_3 = b

Determine all ordered quadruples of real numbers

(x_1, x_2, x_3, x_4)

for which the following system of equations exists, is fulfilled:

x_1 + ax_2 + x_3 = b

x_2 + ax_3 + x_4 = b

x_3 + ax_4 + x_1 = b

x_4 + ax_1 + x_2 = b

Here

a

and

b

are real numbers (case distinction!).

2

1

min area a plane cuts a cube

Which of all planes, the one and the same body diagonal of a cube with the edge length

a

, cuts out a cut figure with the smallest area from the cube? Calculate the area of such a cut figure.
[hide=original wording]Welche von allen Ebenen, die eine und dieselbe Korperdiagonale eines Wurfels mit der Kantenlange a enthalten, schneiden aus den W¨urfel eine Schnittfigur kleinsten Flacheninhaltes heraus? Berechnen Sie den Fl¨acheninhalt solch einer Schnittfigur!

6

1

isosceles iff 2 equal angle bisectors: Germany 1998 R3 9.3. 1968 Ρ4 12.6

Prove the following two statements: (a) If a triangle is isosceles, then two of its bisectors are of equal length. (b) If two angle bisectors in a triangle are of equal length, then it is isosceles.

1968 German National Olympiad

Subcontests

sin x +1/2 sin 2x +1/3 sin 3x > 0

digits of 16 naturals form a geometric sequence

a f(x^n) + f(-x^n) = b

4x4 system, x_1 + ax_2 + x_3 = b

min area a plane cuts a cube

isosceles iff 2 equal angle bisectors: Germany 1998 R3 9.3. 1968 Ρ4 12.6

1968 German National Olympiad

Subcontests

sin x +1/2 sin 2x +1/3 sin 3x &gt; 0

digits of 16 naturals form a geometric sequence

a f(x^n) + f(-x^n) = b

4x4 system, x_1 + ax_2 + x_3 = b

min area a plane cuts a cube

isosceles iff 2 equal angle bisectors: Germany 1998 R3 9.3. 1968 &Rho;4 12.6

sin x +1/2 sin 2x +1/3 sin 3x > 0

isosceles iff 2 equal angle bisectors: Germany 1998 R3 9.3. 1968 Ρ4 12.6