3

Part of 1993 Hungary-Israel Binational

Problems(2)

five points on a semicircle with radius 1

Source: 4-th Hungary-Israel Binational Mathematical Competition 1993

Distinct points

A, B , C, D, E

are given in this order on a semicircle with radius

1

AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.

inequalitiestrigonometrysymmetryfunctionvectorinequalities unsolved

elements of n-th symmetric group

Source: 4-th Hungary-Israel Binational Mathematical Competition 1993

In the questions below:

G

is a ﬁnite group;

H \leq G

G; |G : H |

H

G; |X |

the number of elements of

X \subseteq G; Z (G)

G; G'

the commutator subgroup of

G; N_{G}(H )

the normalizer of

H

G; C_{G}(H )

the centralizer of

H

G

S_{n}

n

-th symmetric group. Show that every element of

S_{n}

is a product of

2

group theoryabstract algebrafunctionsuperior algebrasuperior algebra unsolved