SCHWEITZER 1963

Show that the perimeter of an arbitrary planar section of a tetrahedron is less than the perimeter of one of the faces of the tetrahedron. [Gy. Hajos]

Show that the center of gravity of a convex region in the plane halves at least three chords of the region. [Gy. Hajos]

Let $R=R_1\oplus R_2$ be the direct sum of the rings $R_1$ and $R_2$ , and let $N_2$ be the annihilator ideal of $R_2$ (in $R_2$ ). Prove that $R_1$ will be an ideal in every ring $\widetilde{R}$ containing $R$ as an ideal if and only if the only homomorphism from $R_1$ to $N_2$ is the zero homomorphism. [Gy. Hajos]

Call a polynomial positive reducible if it can be written as a product of two nonconstant polynomials with positive real coefficients. Let $f(x)$ be a polynomial with $f(0)\not=0$ such that $f(x^n)$ is positive reducible for some natural number $n$ . Prove that $f(x)$ itself is positive reducible. [L. Redei]

Let $H$ be a set of real numbers that does not consist of $0$ alone and is closed under addition. Further, let $f(x)$ be a real-valued function defined on $H$ and satisfying the following conditions: $$f(x)\leq f(y)\ \mathrm{if} x \leq y$$ and $$f(x+y)=f(x)+f(y) \;(x,y \in H)\ .$$ Prove that $f(x)=cx$ on $H$ , where $c$ is a nonnegative number. [M. Hosszu, R. Borges]

Show that if $f(x)$ is a real-valued, continuous function on the half-line $0\leq x < \infty$ , and $$\int_0^{\infty} f^2(x)dx <\infty$$ then the function $$g(x)=f(x)-2e^{-x}\int_0^x e^tf(t)dt$$ satisfies $$\int _0^{\infty}g^2(x)dx=\int_0^{\infty}f^2(x)dx.$$ [B. Szokefalvi-Nagy]

Prove that for every convex function $f(x)$ defined on the interval $-1\leq x \leq 1$ and having absolute value at most $1$ , there is a linear function $h(x)$ such that $$\int_{-1}^1|f(x)-h(x)|dx\leq 4-\sqrt{8}.$$ [L. Fejes-Toth]

Let the Fourier series $$\frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)$$ of a function $f(x)$ be absolutely convergent, and let $$a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .$$ Show that $$\frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)$$ is uniformly bounded in $h$ . [K. Tandori]

Let $f(t)$ be a continuous function on the interval $0 \leq t \leq 1$ , and define the two sets of points $$A_t=\{(t,0): t\in[0,1]\} , B_t=\{(f(t),1): t\in [0,1]\}.$$ Show that the union of all segments $\overline{A_tB_t}$ is Lebesgue-measurable, and find the minimum of its measure with respect to all functions $f$ . [A. Csaszar]

Select $n$ points on a circle independently with uniform distribution. Let $P_n$ be the probability that the center of the circle is in the interior of the convex hull of these $n$ points. Calculate the probabilities $P_3$ and $P_4$ . [A. Renyi]