Herzog Competition - 1981

The Ninth Annual Herzog Prize Examination

November 7, 1981

Problem 1:

Does the inequality

{(2 n)}^{n} + {(2 n + 1)}^{n} \geq {(2 n + 2)}^{n}

hold for all positive integers n?

Problem 2:

(L.M. Kelly) A point P moves on the positive x-axis and point Q on the positive y-axis so that the line PQ is always tangent to the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 .

Prove that when the triangle POQ has minimum area the point of tangency will bisect the segment PQ.

(Here O is the origin.)

Problem 3:

(J. Marik) Let f,g be nondecreasing on [0,1]. Prove that

\int_{0}^{1} f (x) dx \int_{0}^{1} g (x) dx \leq \int_{0}^{1} f (x) g (x) dx .

Problem 4:

(L.M. Kelly) If a,b,c are real and $b^{2} < 2 a c$ , prove that the cubic equation

x^{3} + a x^{2} + b x + c = 0

cannot have all real roots.

Problem 5:

Prove

\sin 10 \sin 30 \sin 50 \sin 70 = \frac{1}{16} .

Problem 6:

Three points are taken at random on a unit sphere. What is the probability that the area of the spherical triangle exceeds the area of a great circle?