Herzog Competition - 2011

HERZOG MATH COMPETITION

November 13, 2011

Note: The grading will be similar to the grading of Putnam. This means you must explain your answers/steps as much as possible. An answer without any explanation will receive no credit.

Problem 1

While working on a problem, a student didn’t notice the multiplication sign between two three-digit numbers, so she ended up writing a six digit number instead. The six digit number was seven times bigger than the product of the two three-digit numbers. Can you find the two three-digit numbers?

Problem 2

Define polynomials $f_{n} (x)$ for $n \geq 0$ by $f_{0} (x) = 1$ , $f_{n} (0) = 0$ for $n \geq 1$ , and

\frac{d}{d x} f_{n + 1} (x) = (n + 1) f_{n} (x + 1)

for $n \geq 0$ . Find, with proof, the explicit factorization of $f_{100} (1)$ into powers of distinct primes.

Problem 3

Evaluate

\int_{2}^{4} \frac{\sqrt{\ln (9 - x)} d x}{\sqrt{\ln (9 - x)} + \sqrt{\ln (x + 3)}}

Problem 4

1981 points lie inside a cube of side length 9. Prove that there are two points within a distance less than 1.

Problem 5

Prove that there are no primes in the infinite sequence of integers

10001, 100010001, 1000100010001, \dots

Problem 6

The triangle ABC is inscribed in a circle. The interior bisectors of the angles A, B anc C meet the circle again at A′, B′ and C′ respectively. Prove that the area of triangle A′B′C′ is greater than or equal to the area of the triangle ABC.