Exercise 91. Determine the largest number of integers that can be extracted from the set $\{1,2, \ldots, 2014\}$ such that the difference between any two of these integers is different from 17.
Solution to Exercise J1. This exercise actually stems from a clever use of the pigeonhole principle. Let $E$ be a subset of $\{1,2, \ldots, 2014\}$ containing only integers whose difference is never equal to 17.

First, for any integer $a$, we denote $S_{a}$ as the set $\{a, a+17\}$, and $T_{a}$ as the set $\{a, a+1, \ldots, a+33\}$: then we know that $E \cap S_{a}$ contains at most one element, regardless of the integer $a$ considered: in other words, $\left|E \cap S_{a}\right| \leq 1$. Since $2014=34 \times 59+8$, we know that

\[
\{1,2, \ldots, 2014\} \subseteq \bigcup_{i=0}^{58} T_{34 i+1} \cup \bigcup_{a=2009}^{2014} S_{a}=\bigcup_{i=0}^{58}\left(\bigcup_{a=34 i+1}^{34 i+17} S_{a}\right) \cup \bigcup_{a=2009}^{2014} S_{a}
\]

so that

\[
|E|=|E \cap\{1,2, \ldots, 2014\}| \leq \sum_{i=0}^{58}\left(\sum_{a=34 i+1}^{34 i+17}\left|E \cap S_{a}\right|\right)+\sum_{a=2009}^{2014}\left|E \cap S_{a}\right|=59 \times 17+8=1011
\]

On the other hand, by choosing exactly $E=\bigcup_{i=0}^{58}\left(\bigcup_{a=34 i+1}^{34 i+17}\{a\}\right) \cup \bigcup_{a=2009}^{2014}\{a\}$, we obtain a subset of $\{1,2, \ldots, 2014\}$ containing only integers whose difference is never equal to 17, and whose cardinality is precisely $|E|=1011$.

Thus, the number sought was indeed 1011.

Exercise I2. Let $P$ be a point outside a circle $\mathcal{C}$. The tangents to $\mathcal{C}$ passing through $P$ touch $\mathcal{C}$ at $A$ and $B$. A line passing through $P$ intersects $\mathcal{C}$ at points $Q$ and $R$. Let $S$ be a point on $\mathcal{C}$ such that $(B S) \parallel (Q R)$. Show that $(S A)$ passes through the midpoint of $[Q R]$.