The Winner is Obvious

Algebra Level 5

Sanjoy and Trevor are the last two contestants in an Integration Bee. They are given an undisclosed number of integrals to solve at the same time and the contests ends whenever one of the contestants is unable to solve a problem. Consider the polynomial $x^3 + 3x^2 + 4x - 8$ . This polynomial has three roots $r,s,t$ . The probability that Sanjoy misses an integral first is equal to the reciprocal of the sum of the cube of the roots of the function, which can be represented as $\frac{A}{B}$ , where A and B are relatively prime. The probability that Trevor misses an integral first is equal to the reciprocal of the sum of the square of the roots of the function, which can be represented as $\frac{X}{Y}$ , where X and Y are relatively prime. Determine the winner of the contest and type the product of the numerator and denominator of the probability that they don't miss any integrals in the answer box.

Hint: Think about symmetric sums and the fact that I am a firm believer that Issac Newton discovered Calculus first.

The answer is 1056.

1 solution

Sanjoy Kundu
Jul 17, 2018

Relevant wiki: Symmetric Polynomials

Based on my hint, consider Newton's Sums. Let $A$ be the sum of the roots of the polynomial, $B$ be the sum of the square of the roots of the polynomial, and $C$ be the sum of the cube of the roots of the polynomial $A + 3 = 0, B + 3A + 8 = 0, C + 3B + 4A - 24 = 0$ . It follows that $A = -3$ so $B = 1$ and $C = 33$ . Since we are asked to find the reciprocal of the latter two values, the probability that Sanjoy misses an integral first is $\frac{1}{33}$ and the probability that Trevor misses an integral first is $\frac{1}{1}$ , hence Trevor obviously loses the contest. Lastly, we are asked to find the complement of the probability that Sanjoy misses an integral first. This is trivially just $1 - 1/33 = 32/33$ . Thus, the product of numerator and denominator associated with this probability is $1056$ .

but how can it be that the probability that sanjay loses exists when trevor misses with probability = 1?

Vishesh Agarwal - 2 years, 2 months ago

The probability that I miss any given integral is 1/33 and the probability that Trevor misses an integral is exactly 1. Note that our probabilities need not add up to 1, only that the probability of us missing any given integral is denoted by the given numerical value. In other words, Trevor is guaranteed to miss any given integral but this does not guarantee that I will solve that same integral. For all intensive purposes, the probabilities are not complements and hence do not need to add up to 1.

Sanjoy Kundu - 2 years ago

The Winner is Obvious

The answer is 1056.

1 solution

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