Question 95
Kevin and Yang are playing a game. Yang has $2017 + \binom{2017}{2}$ cards with their front sides face down on the table. The cards are constructed as follows:
- For each $1 \leq n \leq 2017$, there is a blue card with $n$ written on the back, and a fraction $\frac{a_{n}}{b_{n}}$ written on the front, where $\operatorname{gcd}\left(a_{n}, b_{n}\right) = 1$ and $a_{n}, b_{n} \gt 0$.
- For each $1 \leq i \lt j \leq 2017$, there is a red card with $(i, j)$ written on the back, and a fraction $\frac{a_{i} + a_{j}}{b_{i} + b_{j}}$ written on the front.
It is given no two cards have equal fractions. In a turn Kevin can pick any two cards and Yang tells Kevin which card has the larger fraction on the front. Show that, in fewer than 10000 turns, Kevin can determine which red card has the largest fraction out of all of the red cards.
- For each $1 \leq n \leq 2017$, there is a blue card with $n$ written on the back, and a fraction $\frac{a_{n}}{b_{n}}$ written on the front, where $\operatorname{gcd}\left(a_{n}, b_{n}\right) = 1$ and $a_{n}, b_{n} \gt 0$.
- For each $1 \leq i \lt j \leq 2017$, there is a red card with $(i, j)$ written on the back, and a fraction $\frac{a_{i} + a_{j}}{b_{i} + b_{j}}$ written on the front.
It is given no two cards have equal fractions. In a turn Kevin can pick any two cards and Yang tells Kevin which card has the larger fraction on the front. Show that, in fewer than 10000 turns, Kevin can determine which red card has the largest fraction out of all of the red cards.