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Let a and b be positive real numbers such that both x² + a*x + 2b = 0 and x² + 2b*x + a = 0 have real roots. Which of the following statements is/are TRUE?

1. The number of pairs (a, b) satisfying a² + b² = 1/2023 is 2023
2. The minimum value of a + b is 6
3. Given S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, the number of ordered pairs (a, b) from S such that the two quadratics have exactly one common root is 95
4. The number of ordered pairs (a, b) from S such that both quadratics have both roots identical is 4

Correct answer: The minimum value of a + b is 6

Solution

The conditions a² >= 8b and 4b² >= a define a region. The minimum of a+b occurs at the boundary where a² = 8b and 4b² = a, yielding a = 4 and b = 2, giving a+b = 6.

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