Question

Suppose for any real number \( x \), \( [x] \) denotes the greatest integer less than or equal to \( x \). Let \( L(x, y) = [x] + [y] \) and \( R(x, y) = [2x] + [2y] \). Then is it impossible to find any two positive real numbers \( x \) and \( y \) for which:

• A. \( L(x, y) = R(x, y) \)
• B. \( L(x, y) \neq R(x, y) \)
• C. \( L(x, y)<R(x, y) \)
• D. \( L(x, y)>R(x, y) \)

Hint:

When dealing with greatest integer functions, check the behavior for simple values of \( x \) and \( y \) to explore all conditions.

Answer 1

The Correct Option is D

By evaluating various values of \( x \) and \( y \), we find that it is impossible to have \( L(x, y)>R(x, y) \), which makes the Correct Answer (4).