Kezdőlap


Cím:	Problems of the 1986 Kürschák József Competition
Füzet:	1987/február, 80. oldal	PDF \| MathML
Témakör(ök):	Kürschák József (korábban Eötvös Loránd)

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Problems of the 1986 Kürschák József Competition
1. Prove that three semi lines starting from a given point contain three face diagonals of a cuboid if and only if the semi lines include pairwise acute angles such that their sum is $180^{\circ}$ .
2. Let us assume that $n$ is a positive integral number greater than two. Find the maximum value for $h$ and the minimum value for $H$ such that

\begin{matrix} h < \frac{a_{1}}{a_{1} + a_{2}} + \frac{a_{2}}{a_{2} + a_{3}} + ... + \frac{a_{n}}{a_{n} + a_{1}} < H, \end{matrix}

holds for any positive numbers

a_{1}

a_{2}

...

a_{n}

.
3.

A

and

B

play the following game. They arbitrarily select from among the first

100

positive integral numbers

k

ones. If the sum of the selected numbers is even then

A

wins, if their sum is odd then

B

is the winner. For what values of

k

are equal the chances for

A

and

B