2007 Federal Competition For Advanced Students, Part 1

Part of Austrian MO National Competition

Subcontests

(4)

1

38th Austrian Mathematical Competition 2007

n > 4

be a non-negative integer. Given is the in a circle inscribed convex

n

-gon A_0A_1A_2\dots A_{n \minus{} 1}A_n (A_n \equal{} A_0) where the side A_{i \minus{} 1}A_i \equal{} i (for

1 \le i \le n

). Moreover, let

\phi_i

be the angle between the line A_iA_{i \plus{} 1} and the tangent to the circle in the point

A_i

(where the angle

\phi_i

is less than or equal

90^o

\phi_i

is always the smaller angle of the two angles between the two lines). Determine the sum \Phi \equal{} \sum_{i \equal{} 0}^{n \minus{} 1} \phi_i of these

n

1

38th Austrian Mathematical Competition 2007

Let M(n )\equal{}\{\minus{}1,\minus{}2,\ldots,\minus{}n\}. For every non-empty subset of

M(n )

we consider the product of its elements. How big is the sum over all these products?

1

38th Austrian Mathematical Competition 2007

For every positive integer

n

determine the highest value

C(n)

, such that for every

n

(a_1,a_2,\ldots,a_n)

of pairwise distinct integers (n \plus{} 1)\sum_{j \equal{} 1}^n a_j^2 \minus{} \left(\sum_{j \equal{} 1}^n a_j\right)^2\geq C(n)

1

38th Austrian Mathematical Competition 2007

In a quadratic table with

2007

2007

columns is an odd number written in each field. For

1\leq i\leq2007

Z_i

the sum of the numbers in the

i

-th row and for

1\leq j\leq2007

S_j

the sum of the numbers in the

j

A

is the product of all

Z_i

B

the product of all

S_j

. Show that A\plus{}B\neq0