BG-MO 1983

Determine all natural numbers $n$ for which there exists a permutation $(a_1,a_2,\ldots,a_n)$ of the numbers $0,1,\ldots,n-1$ such that, if $b_i$ is the remainder of $a_1a_2\cdots a_i$ upon division by $n$ for $i=1,\ldots,n$ , then $(b_1,b_2,\ldots,b_n)$ is also a permutation of $0,1,\ldots,n-1$ .

Let $b_1\ge b_2\ge\ldots\ge b_n$ be nonnegative numbers, and $(a_1,a_2,\ldots,a_n)$ be an arbitrary permutation of these numbers. Prove that for every $t\ge0$ , $$(a_1a_2+t)(a_3a_4+t)\cdots(a_{2n-1}a_{2n}+t)\le(b_1b_2+t)(b_3b_4+t)\cdots(b_{2n-1}b_{2n}+t).$$

A regular triangular pyramid $ABCD$ with the base side $AB=a$ and the lateral edge $AD=b$ is given. Let $M$ and $N$ be the midpoints of $AB$ and $CD$ respectively. A line $\alpha$ through $MN$ intersects the edges $AD$ and $BC$ at $P$ and $Q$ , respectively.

(a) Prove that $AP/AD=BQ/BC$ . (b) Find the ratio $AP/AD$ which minimizes the area of $MQNP$ .

Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point.