Herzog Competition - 2016

Problem 1

The numbers $Structured MathML transcription$ are in arithmetic progression, and the numbers $Structured MathML transcription$ are in geometric progression. We know that $Structured MathML transcription$, $Structured MathML transcription$, $Structured MathML transcription$, and $Structured MathML transcription$. Find $Structured MathML transcription$ and $Structured MathML transcription$.

Problem 2

Determine $Structured MathML transcription$ so that the equation in $Structured MathML transcription$

Problem 3

Little Marguerite is playing with little Inés, who became very good at multiplications and such (since the Herzog competition of 2014), and wants her to guess a number she wrote in a paper. So Marguerite tells Inés that the number $Structured MathML transcription$ is the smallest natural number $Structured MathML transcription$ which satisfies that

• its decimal representation has a 6 as its last digit, and
• if the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $Structured MathML transcription$.

How can little Inés find the number (only using mathematics and not cheating, of course)?

Problem 4

Given a circle of radius 1 and $Structured MathML transcription$ points $Structured MathML transcription$ in the plane, prove that there is a point $Structured MathML transcription$ on the circle such that $Structured MathML transcription$.

Problem 5

A rectangle in the plane with sides parallel to the $Structured MathML transcription$ and $Structured MathML transcription$ axes is partitioned into several smaller rectangles (which also have sides parallel to the axes). Each of the smaller rectangles has at least one side of integer length. Prove that the big rectangle has at least one side of integer length. (Note: “partition” means that the bigger rectangle is “cut” into the smaller rectangles, so that the union of all the smaller rectangles is the big rectangle, and the smaller rectangles do not overlap except possibly for touching sides or vertices.)

Problem 6

Evaluate $Structured MathML transcription$, where $Structured MathML transcription$ for $Structured MathML transcription$ denotes the number $Structured MathML transcription$ in the interval $Structured MathML transcription$ with $Structured MathML transcription$.