Let \( p(x) \) be a polynomial. When \( p(x) \) is divided by \( (x - 1) \), it leaves 2 as the remainder. When \( p(x) \) is divided by \( (x - 2) \), it leaves 1 as the remainder. What is the remainder when \( p(x) \) is divided by \( (x - 1)(x - 2) \)?
Consider the following in respect of a positive real number \( x \):
I. \( x + \frac{1}{x} > 1 \)
II. \( \left(x + \frac{1}{x}\right)^{2} > 2 \)
III. \( \left(x + \frac{1}{x}\right)^{4} > 9 \)
Which of the above are correct?
Let \( x \) and \( y \) be natural numbers, each less than 20, such that \( x \), \( y \), \( x + y \) and \( x - y \) are prime numbers. How many such combinations of \( (x, y, x + y, x - y) \) are possible?
