To determine the valid declarations, let's go through each option:
Option A) int $x - This option is a valid declaration. In programming languages like C or C++, variable names can start with an underscore (_) or a letter. Therefore, the variable name "$x" is valid.
Option B) int 123 - This option is not a valid declaration. Variable names cannot start with a number.
Option C) int 123 - This option is a valid declaration. Variable names can start with an underscore (), so "_123" is a valid variable name.
Option D) int #dim - This option is not a valid declaration. Variable names cannot start with special characters like "#" in most programming languages.
Option E) int %percent - This option is not a valid declaration. Variable names cannot start with special characters like "%" in most programming languages.
Option F) int central_sales_region_Summer_2005_gross_sales - This option is a valid declaration. Variable names can contain letters, numbers, and underscores. Therefore, "central_sales_region_Summer_2005_gross_sales" is a valid variable name.
The correct options are A, C, and F.