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 $2|AD| = |ED|$. [asy][asy] unitsize(1 cm); pair A, B, C, D, E; A = (0,0); B = (2,0); C = (1.8,1.8); D = extension(C, incenter(A,B,C), A, B); E = extension((C + D)/2, (C + D)/2 + rotate(90)*(C - D), A, B); draw((E + (0.5,0))--A--C--B); draw(C--D); draw(interp((C + D)/2,E,-0.3)--interp((C + D)/2,E,1.2)); dot("$A$", A, SW); dot("$B$", B, S); dot("$C$", C, N); dot("$D$", D, S); dot("$E$", E, S); [/asy][/asy]

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?