Herzog Competition - 1994

Problem 1

Find the sum of all the five-digit numbers that can be formed by arranging all of the five digits 1, 2, 3, 4, 5.

Problem 2

Let $f\left(x\right)=x^3-3x+2$. Suppose that the cubic curve $y=f\left(x\right)$ is cut by a non-vertical straight line in the points A(a, f(a)), B(b, f(b)) and C(c, f(c)). Find (with proof) the coordinates of C in terms of a and b.

Problem 3

Find (with proof) the smallest constant K such that

for all positive numbers x, y.

Problem 4

Prove: For every positive integer n, there is a positive integer $M\left(n\right)$ containing only 1’s and 2’s in its decimal representation, and $M\left(n\right)$ is divisible by $2^n$.

Problem 5

Find a closed form expression for

Problem 6

Let $P_1,P_2,\dots$ be an infinite set of points on the x-axis with integer coordinates, and let Q be an arbitrary point in the plane not on the x-axis. Prove that infinitely many of the distances $\left|P_{n}Q\right|$ are not integers.