SCHWEITZER 1953

Let $a_{v}$ and $b_{v}$ , ${v= 1,2,\dots,n}$ , be real numbers such that $a_{1}\geq a_{2} \geq a_{3}\geq\dots\geq a_{n}> 0$ and $b_{1}\geq a_{1}, b_{1}b_{2}\geq a_{1}a_{2},\dots,b_{1}b_{2}\dots b_{n}\geq a_{1}a_{2}\dots a_{n}$ Show that $b_{1}+b_{2}+\dots+b_{n}\geq a_{1}+a_{2}+\dots+a_{n}$ (S. 4)

Place 32 white and 32 black chessmen on the chessboard. Two chessmen of different colours will be said to form a "related pair" if they are placed either in the same row or in the same column. Determine the maximum and minimum number of related pairs (over all possible arrangements of the 64 chessmen considered. (C. 2)

Denoting by $E$ the class of trigonometric polynomials of the form $f(x)=c_{0}+c_{1}cos(x)+\dots +c_{n} cos(nx)$ , where $c_{0} \geq c_{1} \geq \dots \geq c_{n}>0$ , prove that $(1-\frac{2}{\pi})\frac{1}{n+1}\leq min_{{f\epsilon E}}( \frac{max_{\frac{\pi}{2}\leq x\leq \pi} \left | f(x) \right |}{max_{0\leq x\leq 2\pi} \left | f(x) \right |})\leq (\frac{1}{2}+\frac{1}{\sqrt{2}})\frac{1}{n+1}$ . (S. 24)

Show that every closed curve c of length less than $2\pi$ on the surface of the unit sphere lies entirely on the surface of some hemisphere of the unit sphere. (G. 8)

Show that any positive integer has at least as many positive divisors of the form $3k+1$ as of the form $3k-1$ . (N. 7)

Let $H_{n}(x)$ be the nth Hermite polynomial. Find $\lim_{n \to \infty } (\frac{y}{2n})^{n} H_{n}(\frac{n}{y})$ For an arbitrary real y. (S.5) $H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{{-x^2}}\right)$

Consider four real numbers $t_{1},t_{2},t_{3},t_{4}$ such that each is less than the sum of the others. Show that there exists a tetrahedron whose faces have areas $t_{1},t_{2}, t_{3}$ and $t_{4},$ respectively. (G. 9)

Does there exist a Euclidean ring which is properly contained in the field $V$ of real numbers, and whose quotient field is $V$ ? (A.21)

Let $w=f(x)$ be regular in $\left | z \right |\leq 1$ . For $0\leq r \leq 1$ , denote by c, the image by $f(z)$ of the circle $\left | z \right | = r$ . Show that if the maximal length of the chords of $c_{1}$ is $1$ , then for every $r$ such that $0\leq r \leq 1$ , the maximal length of the chords of c, is not greater than $r$ . (F. 1)

Consider a point performing a random walk on a planar triangular lattice and suppose that it moves away with equal probability from any lattice point along any one of the six lattice lines issuing from it. Prove that if the walk is continued indefinitely, then the point will return to its starting point with probability 1. (P. 5)