MathDB

1

sum of digits of 99x9999x...x99..99

Calculate the sum of digit of the number

9 \times 99 \times 9999 \times ... \times \underbrace{ 99...99}_{2^n}

where the number of nines doubles in each factor.

9

1

(a^2 + b^2 + 1)(c^2 + d^2 + 1) >= 2(a + c)(b + d)

Let

a, b, c, d

be real numbers. Prove that is

(a^2 + b^2 + 1)(c^2 + d^2 + 1) \ge 2(a + c)(b + d).

10

1

a^n-a is divisible by n , for all integers a

Decide whether there is an integer

n > 1

with the following properties: (a)

n

is not a prime number. (b) For all integers

a

,

a^n - a

is divisible by

n

8

1

people from n different countries sit at a round table

People from n different countries sit at a round table. Assume that for every two members of the same country their neighbours sitting next to them on the right hand side are from different countries. Find the largest possible number of people sitting around the table?

6

1

A won against B, B won against C and C won against A

At least three players have participated in a tennis tournament. Evey two players have played each other exactly once, and each player has at least one match won. Show that there are three players

A,B,C

such that

A

won against

B

,

B

won against

C

and

C

won against

A

.

1

yf(2x) - xf(2y) = 8xy(x^2 - y^2)

Find all functions

f : R \to R

such that for all

x, y \in R

holds

yf(2x) - xf(2y) = 8xy(x^2 - y^2).

4

1

3n points on a circle

A circle with circumference 6n units is given and 3n points divide the circumference in n intervals of 1 unit, n intervals of 2 units, and n intervals of 3 units. Prove that there is at least one pair of points that are diametrically opposite to each other.

2006 Switzerland - Final Round

Subcontests

sum of digits of 99x9999x...x99..99

(a^2 + b^2 + 1)(c^2 + d^2 + 1) >= 2(a + c)(b + d)

a^n-a is divisible by n , for all integers a

people from n different countries sit at a round table

A won against B, B won against C and C won against A

yf(2x) - xf(2y) = 8xy(x^2 - y^2)

3n points on a circle

2006 Switzerland - Final Round

Subcontests

sum of digits of 99x9999x...x99..99

(a^2 + b^2 + 1)(c^2 + d^2 + 1) &gt;= 2(a + c)(b + d)

a^n-a is divisible by n , for all integers a

people from n different countries sit at a round table

A won against B, B won against C and C won against A

yf(2x) - xf(2y) = 8xy(x^2 - y^2)

3n points on a circle

(a^2 + b^2 + 1)(c^2 + d^2 + 1) >= 2(a + c)(b + d)