NL-NWO 2022

A positive integer n is called primary divisor if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$ , $2$ , $4$ , and $8$ each differ by $1$ from a prime number ( $2$ , $3$ , $5$ , and $7$ , respectively), while $9$ is not divisor primary, because the divisor $9$ does not differ by $1$ from a prime number (both $8$ and $10$ are composite). Determine the largest primary divisor number.

A set consisting of at least two distinct positive integers is called centenary if its greatest element is $100$ . We will consider the average of all numbers in a centenary set, which we will call the average of the set. For example, the average of the centenary set $\{1, 2, 20, 100\}$ is $\frac{123}{4}$ and the average of the centenary set $\{74, 90, 100\}$ is $88$ . Determine all integers that can occur as the average of a centenary set.

Given a positive integer $c$ , we construct a sequence of fractions $a_1, a_2, a_3,...$ as follows: $\bullet$ $a_1 =\frac{c}{c+1}$ $\bullet$ to get $a_n$ , we take $a_{n-1}$ (in its most simplified form, with both the numerator and denominator chosen to be positive) and we add $2$ to the numerator and $3$ to the denominator. Then we simplify the result again as much as possible, with positive numerator and denominator. For example, if we take $c = 20$ , then $a_1 =\frac{20}{21}$ and $a_2 =\frac{22}{24} = \frac{11}{12}$ . Then we find that $a_3 =\frac{13}{15}$ (which is already simplified) and $a_4 =\frac{15}{18} =\frac{5}{6}$ . (a) Let $c = 10$ , hence $a_1 =\frac{10}{11}$ . Determine the largest $n$ for which a simplification is needed in the construction of $a_n$ . (b) Let $c = 99$ , hence $a_1 =\frac{99}{100}$ . Determine whether a simplification is needed somewhere in the sequence. (c) Find two values of $c$ for which in the first step of the construction of $a_5$ (before simplification) the numerator and denominator are divisible by $5$ .

In triangle $ABC$ , the point $D$ lies on segment $AB$ such that $CD$ is the angle bisector of angle $\angle C$ . The perpendicular bisector of segment $CD$ intersects the line $AB$ in $E$ . Suppose that $|BE| = 4$ and $|AB| = 5$ . (a) Prove that $\angle BAC = \angle BCE$ . (b) Prove that $2|AD| = |ED|$ .

$A$ $B$ $C$ $D$ $E$

Kira has $3$ blocks with the letter $A$ , $3$ blocks with the letter $B$ , and $3$ blocks with the letter $C$ . She puts these $9$ blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence $ABCAABCBC$ the blocks with the letter A have distances $1, 3$ , and $4$ between one another, the blocks with the letter $B$ have distances $2, 4$ , and $6$ between one another, and the blocks with the letter $C$ have distances $2, 4$ , and $6$ between one another. Altogether, we got distances of $1, 2, 3, 4$ , and $6$ ; these are $5$ distinct distances. What is the maximum number of distinct distances that can occur?