1994 Italy TST

Part of Italy TST

Subcontests

(4)

1

\sum_{(E_1,...,E_k) \in E_k }|E_1 \cap ... \cap E_k|

X

n

k

be a positive integer. Consider the family

S_k

k

(E_1,...,E_k)

E_i \subseteq X

i

. Evaluate the sums

\sum_{(E_1,...,E_k) \in S_k }|E_1 \cap ... \cap E_k|

\sum_{(E_1,...,E_k) \in S_k }|E_1 \cup ... \cup E_k|

1

f(x- f(y)) = 1+x-y

Find all functions

f : R \to R

satisfying the condition

f(x- f(y)) = 1+x-y

x,y \in R

1

(2^{p-1} -1)}/p is a perfect square

Find all prime numbers

p

\frac{2^{p-1} -1}{p}

is a perfect square.

1

min and max of perpendicular chords of circle passing through P

\gamma

P

inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of

\gamma

passing through

P