MathDB

1

periodic are last digit of n^{(n^n)}

a_n

the last digit of the number

n^{(n^n)}

(let

n\ne 0

be a natural number ). Prove that the numbers

a_n

form a periodic sequence and state this period!

2

1

[sin 3x cos ( \pi/ 3 -4x )+ 1 ]/ [sin (\pi / 3-7x) - cos (\pi / 6+x)+m]=0

Find all real values

x

that satisfy the following equation:

\frac{\sin 3x cos \left(\frac{\pi}{3}-4x \right)+ 1}{\sin \left(\frac{\pi}{3}-7x \right) - cos\left(\frac{\pi}{6}+x \right)+m}= 0

where

m

is a given real number.

1

\frac{a + b}{2} \ge \sqrt[a+b]{a^b b^a}

Prove that for all positive, entirely rational numbers

a

and

b

always holds

\frac{a + b}{2} \ge \sqrt[a+b]{a^b \cdot b^a}.

When does the equal sign hold?

6

1

equilateral and equiangular cyclic and tangential polygons

Which of the following four statements are true and which are false? a) If a polygon inscribed in a circle is equilateral, then it is also equiangular. b) If a polygon inscribed in a circle is equiangular, then it is also equilateral. c) If a polygon circumscribed to a circle is equilateral, then it is also equiangular. d) If a polygon circumscribed to a circle is equiangular, then it is also equilateral.

5

1

<C= ? if <B=45^o and BP : PC =1 : 2, APC = 60^o , without trigonometry

A triangle

ABC

with

\beta= 45^o

is given. There is a point

P

on side

BC

, where

BP : PC =1 : 2

(inner division) and

\angle APC = 60^o

. Someone claims that you can do it with elementary geometric theorems alone without using the plane trigonometry, the size of the angle

\gamma

determine.

3

1

centers of insphere and circumsphere coincide, tetrahedron , equal in area sides

Given a (not necessarily regular) tetrahedron, all of its sides are equal in area. Prove that the following points then coincide: a) the center of the inscribed sphere, i.e. all four side surfaces internally touching sphere, b) the center of the surrounding sphere, i.e. the sphere passing through the four vertixes.

1964 German National Olympiad

Subcontests

periodic are last digit of n^{(n^n)}

[sin 3x cos ( \pi/ 3 -4x )+ 1 ]/ [sin (\pi / 3-7x) - cos (\pi / 6+x)+m]=0

\frac{a + b}{2} \ge \sqrt[a+b]{a^b b^a}

equilateral and equiangular cyclic and tangential polygons

<C= ? if <B=45^o and BP : PC =1 : 2, APC = 60^o , without trigonometry

centers of insphere and circumsphere coincide, tetrahedron , equal in area sides

1964 German National Olympiad

Subcontests

periodic are last digit of n^{(n^n)}

[sin 3x cos ( \pi/ 3 -4x )+ 1 ]/ [sin (\pi / 3-7x) - cos (\pi / 6+x)+m]=0

\frac{a + b}{2} \ge \sqrt[a+b]{a^b b^a}

equilateral and equiangular cyclic and tangential polygons

&lt;C= ? if &lt;B=45^o and BP : PC =1 : 2, APC = 60^o , without trigonometry

centers of insphere and circumsphere coincide, tetrahedron , equal in area sides

<C= ? if <B=45^o and BP : PC =1 : 2, APC = 60^o , without trigonometry