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This problem trains for: AMC-10, AIME, AMC-12, Math Kangaroo 9-10.

What real values can the parameter p have in order for the inequality below to be true for any real value of the variable x?

equation

For the inequality:

equation

to be true for any x the quadratic function associated with it must be concave and not have more than one real root.

For the parabola to be concave we must have:

equation

For the parabola to not have two distinct real roots, the discriminant of the equation must be negative or zero:

equation

equation

We now have to solve the inequality:

equation

Solve the associated equation:

equation

The solution of the inequality is:

equation

We have to intersect this set of values with the set obtained from the condition to be concave. Since:

equation

it follows that there are no real values of p for which the conditions can be fulfilled.

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