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IF x^3 + f(y)*x + f(z) = 0 THEN f(x)^3 + y*f(x) + z = 0

Source: German TST 3, P3, 2009, Exam set by Gunther Vogel

7/18/2009

Find all functions

f: \mathbb{R} \mapsto \mathbb{R}

such that

\forall x,y,z \in \mathbb{R}

we have: If x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0, then f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.

functionalgebrapolynomialcalculusGermany

Assume segments AA', BB' and CC' have the same length

Source: German TST 6, P3, 2009

7/18/2009

Let

A,B,C,M

points in the plane and no three of them are on a line. And let

A',B',C'

points such that

MAC'B, MBA'C

and

MCB'A

are parallelograms: (a) Show that \overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} < \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}. (b) Assume segments

AA', BB'

and

CC'

have the same length. Show that 2 \left(\overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} \right) \leq \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}. When do we have equality?

geometryparallelogramgeometry unsolved

At some point 2008 is written on the board

Source: AIMO 3, German Pre-TST 2009

7/16/2011

Initially, on a board there a positive integer. If board contains the number

x,

then we may additionally write the numbers

2x+1

and

\frac{x}{x+2}.

At some point 2008 is written on the board. Prove, that this number was there from the beginning.

invariantnumber theory unsolvednumber theory

What is the highest possible point number

Source: AIMO 6, German Pre-TST 2009

7/16/2011

The 16 fields of a

4 \times 4

checker board can be arranged in 18 lines as follows: the four lines, the four columns, the five diagonals from north west to south east and the five diagonals from north east to south west. These diagonals consists of 2,3 or 4 edge-adjacent fields of same colour; the corner fields of the chess board alone do not form a diagonal. Now, we put a token in 10 of the 16 fields. Each of the 18 lines contains an even number of tokens contains a point. What is the highest possible point number when can be achieved by optimal placing of the 10 tokens. Explain your answer.

combinatorics unsolvedcombinatorics

3

Problems(4)