To determine which declarations are valid, we need to consider the rules for naming variables in programming languages like C++.
A. int $x; - This declaration is valid. In C++, variable names can start with a letter, underscore, or a dollar sign.
B. int 123; - This declaration is invalid. Variable names cannot start with a digit.
C. int _123; - This declaration is valid. Variable names can start with an underscore.
D. int #dim; - This declaration is invalid. Variable names cannot start with a special character like a hash or pound sign.
E. int %percent; - This declaration is invalid. Variable names cannot start with a special character like a percent sign.
F. int central_sales_region_Summer_2005_gross_sales; - This declaration is valid. Variable names can contain letters, numbers, and underscores.
Therefore, the valid declarations are A, C, and F.