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[ { "content": "Question: In triangle $ABC$, let $D$ be a point on $BC$ such that $AD$ bisects $\\angle BAC$. If $AB=10$, $AC=15$, and $BC=12$, find the length of $AD$.\nPlease reason step by step, and put your final answer within \\boxed{}.", "role": "user" }, { "content": "<think>\nOkay, so I need t...
[ { "content": "Question: In a certain game, a player starts with $1 and flips a fair coin. If the coin lands on heads, the player gains $1. If it lands on tails, the player loses $0.50. The game ends when the player either has $3 or $0. What is the probability that the player ends the game with $3?\nPlease reaso...