Herzog Competition - 2015

HERZOG MATH COMPETITION

November 14, 2015

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

Let $Structured MathML transcription$ be an $Structured MathML transcription$ matrix such that $Structured MathML transcription$ for each $Structured MathML transcription$. Prove that $Structured MathML transcription$ is invertible. Here $Structured MathML transcription$ is the $Structured MathML transcription$ identity matrix.

Problem 2

Prove that for any non-negative integer $Structured MathML transcription$, the number

$$Structured MathML transcription$$

is not prime.

Problem 3

Let $Structured MathML transcription$ be a continuous function such that $Structured MathML transcription$ for all $Structured MathML transcription$. Prove that the range of $Structured MathML transcription$ is all of $Structured MathML transcription$.

Problem 4

Consider a circle of diameter AB and center O, and the tangent t at B. A variable tangent to the circle with contact point M intersects t at P. Find the locus of the point Q where the line OM intersects the parallel through P to the line AB.

Problem 5

A coin is tossed $Structured MathML transcription$ times. What is the probability that two heads will turn up in succession somewhere in the sequence?

Problem 6

Show that the number $Structured MathML transcription$ can be written as the sum of four perfect cubes, but not as the sum of three perfect cubes.