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Problem 644

AMC 10/12, early questions

Combinatorics Difficulty 3.0 Prove it UKMT Senior Kangaroo · United Kingdom · 2023

Kiran is designing a game which involves a bag of twenty-one marbles. Some of the marbles are blue, the rest are red. To play the game, two marbles are drawn out. The game is won if at least one red marble is drawn. To ensure the probability of the game being won is exactly one-half, Kiran uses $B$ blue marbles and $R$ red marbles. What is the value of $B^2 + R^2$ ?

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