IL-GILLIS 2018

$n$ people sit in a circle. Each of them is either a liar (always lies) or a truthteller (always tells the truth). Every person knows exactly who speaks the truth and who lies. In their turn, each person says 'the person two seats to my left is a truthteller'. It is known that there's at least one liar and at least one truthteller in the circle.

[list=a] [] Is it possible that $n=2017?$ [] Is it possible that $n=5778?$ [/list]

An arithmetic sequence is an infinite sequence of the form $a_n=a_0+n\cdot d$ with $d\neq 0$ .

A geometric sequence is an infinite sequence of the form $b_n=b_0 \cdot q^n$ where $q\neq 1,0,-1$ .

[list=a] [] Does every arithmetic sequence of integers have an infinite subsequence which is geometric? [] Does every arithmetic sequence of real numbers have an infinite subsequence which is geometric? [/list]

Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.

The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?

The sequence $a_n$ is defined for any $n\geq 10$ by the following inductive rule:

• $a_{10}=5778$
• If $a_n=0$ then $a_{n+1}=0$ .
• If $a_n\neq0$ then $a_{n+1}$ is the number whose base- $(n+1)$ representation equals the base $n$ representation of the number $a_n -1$ . [/list] For example, $a_{11}=5\cdot11^3+7\cdot11^2+7\cdot11^1+7\cdot11^0=7586$ $a_{12}=5\cdot12^3+7\cdot12^2+7\cdot12^1+6\cdot12^0=9738$ [list=a]
• Does there exist $n\geq10$ for which $a_n=0$ ?
• Is $a_{1,000,000}=0$ ?
• Is $a_{100^{100^{100}}}=0$ ?

In the corners of triangle $ABC$ there are three circles with the same radius. Each of them is tangent to two of the triangle's sides. The vertices of triangle $MNK$ lie on different sides of triangle $ABC$ , and each edge of $MNK$ is also tangent to one of the three circles. Likewise, the vertices of triangle $PQR$ lie on different sides of triangle $ABC$ , and each edge of $PQR$ is also tangent to one of the three circles (see picture below). Prove that triangles $MNK,PQR$ have the same inradius.

A uniform covering of the integers $1,2,...,n$ is a finite multiset of subsets of $\{1,2,...,n\}$ , so that each number lies in the same amount of sets from the covering. A covering may contain the same subset multiple times, it must contain at least one subset, and it may contain the empty subset. For example, $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})$ is a uniform covering of $1,2,3,4$ (every number occurs in two sets). The covering containing only the empty set is also uniform (every number occurs in zero sets).

Given two uniform coverings, we define a new uniform covering, their sum (denoted by $\oplus$ ), by adding the sets from both coverings. For example: $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})\oplus(\{1\},\{2\},\{3\},\{4\})=$ $(\{1\},\{1\},\{1\},\{2\},\{3\},\{4\},\{2,3\},\{3,4\},\{2,4\})$ A uniform covering is called non-composite if it's not a sum of two uniform coverings.

Prove that for any $n\geq1$ , there are only finitely many non-composite uniform coverings of $1,2,...,n$ .