Herzog Competition - 1997

THE TWENTY-FIFTH HERZOG PRIZE EXAMINATION

November 8, 1997

Problem 1

Prove or disprove: The next-to-last digit of a positive integral power of 3 is always an even number.

Problem 2

Prove: There are no integers $a, b, c, d$ such that $f (x) = a x^{3} + b x^{2} + c x + d$ is equal to 1 when $x = 19$ and equal to 2 when $x = 97$ .

Problem 3

Two diagrams of circles representing cans packed in a box. Method A has three aligned rows of four cans. Method B uses staggered rows with four cans on the top and bottom and three in the middle. — Method A and Method B can-packing diagrams.

Pictured above are two ways to pack cans of soup in a shallow box. Obviously, method A enables the grocer to pack $n^{2}$ cans of unit diameter into an $n \times n$ box. Find the smallest number $n$ for which method B enables the grocer to pack more than $n^{2}$ cans into an $n \times n$ box.

Problem 4

Let $f$ be a differentiable function such that $f (x) + f^{'} (x) \to 0$ as $x \to \infty$ . Prove that $f (x) \to 0$ as well.

Problem 5

Find all pairs $x, y$ of positive integers satisfying $x^{3} - y^{3} = x y + 61$ .

Problem 6

Let $x$ and $y$ be a pair of positive real numbers satisfying $x^{3} + y^{3} = x - y$ .

a.) Prove that infinitely many such pairs of numbers exist.

b.) Prove that for any such pair, $x^{2} + y^{2} < 1$ .