What is the product of all real numbers that are doubled when added to their reciprocals?
This problem is asking for the product of all real numbers x that satisfy the equation 2x = x + 1/x. To solve for x, I can multiply both sides by x to get rid of the fraction: 2x^2 = x^2 + 1. Then I can subtract x^2 from both sides to get x^2 = 1. Taking the square root of both sides, I get x = ±1. So there are only two real numbers that are doubled when added to their reciprocals: 1 and -1. To find their product, I just multiply them together: 1 × (-1) = -1.

