DK-GEORGMOHR 2022

The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$ , the height of the liquid is $5$ cm. a) What is the height of the liquid when it lies as shown in figure $2$ ? b) What is the height of the liquid when it lies as shown in figure $3$ ?

A positive integer is a palindrome if it is written identically forwards and backwards. For example, $285582$ is a palindrome. A six digit number $ABCDEF$ , where $A, B, C, D, E, F$ are digits, is called cozy if $AB$ divides $CD$ and $CD$ divides $EF$ . For example, $164896$ is cozy. Determine all cozy palindromes.

The square $ABCD$ has side length $1$ . The point $E$ lies on the side $CD$ . The line through $A$ and $E$ intersects the line through $B$ and $C$ at the point $F$ . Prove that $$\frac{1}{|AE|^2}+\frac{1}{|AF|^2}= 1.$$

Georg plays the following game. He chooses two positive integers $n$ and $k$ . On an $n \times n$ - board where all the tiles are white, Georg paints k of the tiles black. Then he counts the number of black tiles in each row, forms the square of each of these n numbers and adds up the squares. He calls the result $S$ . In the same way he counts the number of white tiles in each row, forms the square of each of these n numbers and adds up those squares. He calls the result $H$ . Georg would like to achieve $S - H = 49$ . Determine all possible values of n and k for which this is possible.

Example: If Georg chooses $n = 5$ and $k = 14$ , he could for example paint the board as shown. Then $S = 1^2 + 2^2 + 3^2 + 3^2 + 5^2 = 1 + 4 + 9 + 9 + 25 = 48$ , $H = 4^2 + 3^2 + 2^2 + 2^2 + 0^2 = 16 + 9 + 4 + 4 + 0 = 33$ , so in this case $S - H = 48 - 33 = 15$ .

Let $n > 2$ be an integer. The numbers $1, 2, . . . , n$ are written at the vertices of an $n$ -gon in that order. A move consists of choosing two adjacent vertices and adding $1$ to the numbers written there. Determine all n for which it is possible to achieve that all numbers are identical after a finite number of moves.