SCHWEITZER 1955

Let $a_{1}, a_{2}, \dots , a_{n}$ and $b_{1}, b_{2}, \dots , b_{m}$ be $n+m$ unit vectors in the $r$ -dimensional Euclidean space $E_{r} (n,m \leq r)$ ; let $a_{1}, a_{2}, \dots , a_{n}$ as well as $b_{1}, b_{2}, \dots , b_{m}$ be mutually orthogonal. For any vector $x \in E_{r}$ , consider $Tx= \sum_{i=1}^{n}\sum_{k=1}^{m}(x,a_{i})(a_{i},b_{k})b_{k}$ ( $(a,b)$ denotes the scalar product of $a$ and $b$ ). Show that the sequence $(T^{k}x)^{\infty}_{ k =0}$ , where $T^{0} x= x$ and $T^{k} x = T(T^{k-1}x)$ , is convergent and give a geometrical characterization of how the limit depends on $x$ . (S. 14)

Let $f_{1}(x), \dots , f_{n}(x)$ be Lebesgue integrable functions on $[0,1]$ , with $\int_{0}^{1}f_{1}(x) dx= 0$ $(i=1,\dots ,n)$ . Show that, for every $\alpha \in (0,1)$ , there existis a subset $E$ of $[0,1]$ with measure $\alpha$ , such that $\int_{E}f_{i}(x)dx=0$ . (R. 17)

Let the density function $f(x)$ of the random variable $\xi$ bean even function; let further $f(x)$ be monotonically non-increasing for $x>0$ . Suppose that $D^{2}= \int_{-\infty }^{\infty }x^{2}f(x) dx$ exists. Prove that for every non negative $\lambda$ $P(\left |\xi \right |\geq \lambda D)\leq \frac{1}{1+\lambda ^{2}}$ . (P. 7)

Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ( $\alpha , \beta , p, q, r$ need not necessarily be different). (N. 12)

Show that a ring $R$ is commutative if for every $x \in R$ the element $x^{2}-x$ belongs to the centre of $R$ . (A. 18)

For a prime factorisation of a positive integer $N$ let us call the exponent of a prime $p$ the integer $k$ for which $p^{k} \mid N$ but $p^{k+1} \nmid N$ ; let, further, the power $p^{k}$ be called the "contribution" of $p$ to $N$ . Show that for any positive integer $n$ and for any primes $p$ and $q$ the contibution of $p$ to $n!$ is greater than the contribution of $q$ if and only if the exponent of $p$ is greater than that of $q$ .

Prove that for any odd prime number $p$ , the polynomial $2(1+x^{ \frac{p+1}{2} }+(1-x)^{\frac {p+1}{2}})$ is congruent mod $p$ to the square of a polynomial with integer coefficients. (N. 21)

*This problem was proposed by P. Erdõs in the American Mathematical Monthly 53 (1946), p. 594

Show that on any tetrahedron there can be found three acute bihedral angles such that the faces including these angles count among them all faces of tetrahedron. (G. 10)

Show that to any elliptic paraboloid $\varphi_1$ there may be found an elliptic paraboloid $\varphi_2$ (other than $\varphi_1$ ) and an affinity $\phi$ which maps $\varphi_1$ onto $\varphi_2$ and has the following property: If $P$ is any point of $\varphi_1$ such that $\phi(P) \neq P$ , then the straight line connecting $P$ and $\phi(P)$ is a common tangent of the two paraboloids. (G. 18)

Show that if a convex polyhedron has vertices of regular distribution and congruent faces, then it is regular. (A system of points is said to be of regular distribution if every point of the system can be transformed into any other point by congruent transformations mapping the system onto itself.) (G. 11)