Herzog Competition - 2004

Herzog Competition

2004

Problem 1

Let n be a positive integer. Prove that the sum of the digits of ${1981}^n$ is at least 19.

Problem 2

Let A be a square matrix with real entries such that $A^3=A+I$. Show that $\det A\>0$.

Problem 3

You have a chess board of size 10 × 10. Prove that there are no more than 250 ways to tile the board with dominoes (i.e., rectangles of size 2 × 2).

Problem 4

Let f be a real continuous function on [−1, 1] such that |f(t)| ≤ 1 for t ∈ [−1, 1],

$${\int }_{-1}^{1}f(t)\mathrm{dt}=1,{\int }_{-1}^1{f}^2(t)\mathrm{dt}=1$$

Show that

(a) $${\int }_{-1}^1{f}^3(t)\mathrm{dt}\ge 0$$

(b) $${\int }_{-1}^1{f}^3(t)\mathrm{dt}\ge \frac{1}{3}$$

Problem 5

A part of a square of size 1 × 1 is painted in red. It is known that no two red points of the square are at the distance ε (where ε > 0). Prove that

(a) the area of the red part of the square is at most $\frac{1}{3}{(1+\epsilon )}^2$.

(b) the area of the red part of the square is at most $\frac{7}{2}{(1+\sqrt{3}\epsilon )}^2$.