1974 Kurschak Competition

Part of Kürschák Math Competition

Subcontests

(3)

1

polynomial by sum (-x)^{2k}/(2k)! >=0

p_k(x) = 1 -x + \frac{x^2}{2! } - \frac{x^3}{3!}+ ... + \frac{(-x)^{2k}}{(2k)!}

Show that it is non-negative for all real

x

and all positive integers

k

1

squares S_i of side 1/i can be put into a square side k without overlapping.

S_n

is a square side

\frac{1}{n}

. Find the smallest

k

such that the squares

S_1, S_2,S_3, ...

can be put into a square side

k

without overlapping.

1

a library has one exit and one entrance and a blackboard at each

A library has one exit and one entrance and a blackboard at each. Only one person enters or leaves at a time. As he does so he records the number of people found/remaining in the library on the blackboard. Prove that at the end of the day exactly the same numbers will be found on the two blackboards (possibly in a different order).