SCHWEITZER 1964

Among all possible representations of the positive integer $n$ as $n=\sum_{i=1}^k a_i$ with positive integers $k, a_1 < a_2 < ...<a_k$ , when will the product $\prod_{i=1}^k a_i$ be maximum?

Let $p$ be a prime and let $$l_k(x,y)=a_kx+b_ky \;(k=1,2,...,p^2)\ .$$ be homogeneous linear polynomials with integral coefficients. Suppose that for every pair $(\xi,\eta)$ of integers, not both divisible by $p$ , the values $l_k(\xi,\eta), 1\leq k\leq p^2$ , represent every residue class $\textrm{mod} p$ exactly $p$ times. Prove that the set of pairs $\{(a_k,b_k): 1\leq k \leq p^2 \}$ is identical $\textrm{mod} p$ with the set $\{(m,n): 0\leq m,n \leq p-1 \}.$

Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.

Let $A_1,A_2,...,A_n$ be the vertices of a closed convex $n$ -gon $K$ numbered consecutively. Show that at least $n-3$ vertices $A_i$ have the property that the reflection of $A_i$ with respect to the midpoint of $A_{i-1}A_{i+1}$ is contained in $K$ . (Indices are meant $\textrm{mod} n\ .$ )

Is it true that on any surface homeomorphic to an open disc there exist two congruent curves homeomorphic to a circle?

Let $y_1(x)$ be an arbitrary, continuous, positive function on $[0,A]$ , where $A$ is an arbitrary positive number. Let $$y_{n+1}=2 \int_0^x \sqrt{y_n(t)}dt \;(n=1,2,...)\ .$$ Prove that the functions $y_n(x)$ converge to the function $y=x^2$ uniformly on $[0,A]$ .

Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution $f(t)$ and any real number $c,f(t+c)$ is also a solution.

Let $F$ be a closed set in the $n$ -dimensional Euclidean space. Construct a function that is $0$ on $F$ , positive outside $F$ , and whose partial derivatives all exist.

Let $E$ be the set of all real functions on $I=[0,1]$ . Prove that one cannot define a topology on $E$ in which $f_n\rightarrow f$ holds if and only if $f_n$ converges to $f$ almost everywhere.

Let $\varepsilon_1,\varepsilon_2,...,\varepsilon_{2n}$ be independent random variables such that $P(\varepsilon_i=1)=P(\varepsilon_i=-1)=\frac 12$ for all $i$ , and define $S_k=\sum_{i=1}^k \varepsilon_i, 1\leq k \leq 2n$ . Let $N_{2n}$ denote the number of integers $k\in [2,2n]$ such that either $S_k>0$ , or $S_k=0$ and $S_{k-1}>0$ . Compute the variance of $N_{2n}$ .