Herzog Competition - 2000

THE TWENTY-EIGHTH HERZOG PRIZE EXAMINATION

November 4, 2000

Problem 1

The recursion formula $f(n+2)=f(n+1)+f\left(n\right)$ defines the Fibonacci sequence in case we start with the conditions f(0) = f(1) = 1. Of course, if we start with other integral values for f(0) and f(1), different sequences will result. Some of these sequences will contain a multiple of 5 (for example 1,1,2,3,5,… or 8,13,21,34,55,…). However, the starting values f(0) = 1, f(1) = 3 result in the sequence 1,3,4,7,11,18,29,…, and it’s not immediately clear whether a multiple of 5 will ever make an appearance.

Determine how many terms should be calculated in order to ascertain whether or not such a sequence will contain a multiple of 5 for arbitrary starting values.

Problem 2

Determine the locus of all points P(x,y) that satisfy

$$\sqrt{{(x-4)}^2+y^2}-\sqrt{x^2+{(y+3)}^2}=7$$

Problem 3

Can a plane circle pass through exactly three points with rational coordinates?

Problem 4

Let p and q be twin primes greater than 3. Show that $p^2+q^2-2$ is divisible by 72.

Problem 5

Each number in the triple {112,518,322} is divisible by 14. Furthermore, the determinant

$$\left|\begin{array}{ccc}1& 1& 2\\ 5& 1& 8\\ 3& 2& 2\end{array}\right|=14$$

Prove the general result: Any common factor of n arbitrary n-digit numbers is also a factor of the analogously formed n×n determinant.

Problem 6

Let c>1, and define $F(t)={\displaystyle \underset{1}{\overset{c}{\int }}}\frac{\mathrm{d}x}{(x^2+t)}$. Prove that F(t) is continuous at t = 0.