2024 SSMO Relay Round 5 Problems/Problem 2

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Problem

Let $T = TNYWR.$ In the game of high and low, the computer chooses two integers without replacement from the set $\{1,2,3,\dots,T\}$. The computer displays the first integer and asks the player if the second integer is higher or lower. Given that the player always plays optimally, the chances of guessing correctly is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution