## Euclid 2023 Question 7, CEMC UWaterloo

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#### Difficulty: 8

#### This problem is tagged with euclid, euclid23, highschool.

**(Euclid 2023, Question 7, CEMC - UWaterloo)**

(a) A bag contains $3$ red marbles and $6$ blue marbles. Akshan removes one marble at a time until the bag is empty. Each marble that they remove is chosen randomly from the remaining marbles. Given that the first marble that Akshan removes is red and the third marble that they remove is blue, what is the probability that the last two marbles that Akshan removes are both blue?

(b) Determine the number of quadruples of positive integers $(a, b, c, d)$ with $a < b < c < d$ that satisfy both of the following system of equations:

$ac + ad + bc + bd = 2023$

$a + b + c + d = 296$

*Answer Submission Note(s)*

Format fractional answers as "m/n" and separate your answers with a single space.

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