answer:1. Given the perimeter of square A is 20 units, the side length (since the perimeter of a square = 4 times the side length) is: [ text{Side length of A} = frac{20}{4} = 5 text{ units} ] 2. Given the perimeter of square B is 40 units, the side length is: [ text{Side length of B} = frac{40}{4} = 10 text{ units} ] 3. Assuming the side length of square C increases proportionally from square B as square B did from A (doubles), then: [ text{Side length of C} = 2 times 10 = 20 text{ units} ] 4. The area of square A = side length squared = 5^2 = 25 square units, and the area of square C = side length squared = 20^2 = 400 square units. 5. The ratio of the area of square A to square C is: [ frac{text{Area of A}}{text{Area of C}} = frac{25}{400} = boxed{frac{1}{16}} ]

answer:By the Law of Sines, we know that frac {a}{sin A} = frac {b}{sin B} Therefore, sin B = b cdot frac {sin A}{a} = sqrt {3} times frac {1}{2} = frac {sqrt {3}}{2} Since 0 < B < 180^circ, Therefore, B = 60^circ or 120^circ Thus, if a=1, b= sqrt {3}, and A=30^circ, then B=60^circ or 120^circ angle B = 60^circ cannot deduce a=1, b= sqrt {3}, and A=30^circ Hence, the correct choice is boxed{D} Using the Law of Sines to find the value of sin B = b cdot frac {sin A}{a} and determine the relationship between the two propositions. This question tests the Law of Sines and the concept of necessary and sufficient conditions, which is a basic problem that should be mastered.

answer:1. Since f(x+y) = f(x) + f(y) holds for any x, y in mathbb{R}, let x = y = 0, then we have f(0) = f(0) + f(0). Solving this, we get f(0) = 0. Therefore, the value of f(0) is boxed{0}. 2. The function f(x) is an odd function on mathbb{R}. **Proof**: Let y = -x, then f(0) = f(x) + f(-x) = 0. Therefore, f(-x) = -f(x), which means the function f(x) is an odd function on mathbb{R}. Hence, the conclusion is that f(x) is an odd function, which is boxed{text{odd function}}.

answer:1. **Problem Translation and Initial Observations:** We are given ( x_{i} in {sqrt{2}-1, sqrt{2}+1} ) for ( i = 1, 2, 3, ldots, 2012 ), and we need to find how many different positive integer values the sum: [ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + cdots + x_{2001} x_{2002} + x_{2011} x_{2012} ] can attain. 2. **Calculate Possible Products:** We should first calculate the products ( x_{i} x_{i+1} ) for any two terms from the set ( {sqrt{2}-1, sqrt{2}+1} ): - ((sqrt{2}-1)^{2} = 3 - 2sqrt{2}) - ((sqrt{2}+1)^{2} = 3 + 2sqrt{2}) - ((sqrt{2}-1)(sqrt{2}+1) = sqrt{2}^{2} - 1 = 2 - 1 = 1) 3. **Count the Number of Terms:** There are 1006 such pairs because there are 2012 elements and each pair consists of two elements. 4. **Define Variables and System of Equations:** Let: - ( a ) be the number of products valued ( 3 - 2sqrt{2} ), - ( b ) be the number of products valued ( 3 + 2sqrt{2} ), - ( c ) be the number of products valued 1. Then, the equation that balances the number of terms is: [ a + b + c = 1006 ] 5. **Expression for S:** The sum ( S ) can then be written as: [ S = a(3 - 2sqrt{2}) + b(3 + 2sqrt{2}) + c ] 6. **Separate Real and Imaginary Parts:** Separating the real and the irrational parts, we get: [ S = 3a + 3b + c + 2sqrt{2}(b - a) ] 7. **Condition for S to be an Integer:** For ( S ) to be a positive integer, the irrational part must be zero. Therefore: [ b - a = 0 implies b = a ] Substitute ( b = a ) into the equation ( a + b + c = 1006 ): [ a + a + c = 1006 implies 2a + c = 1006 ] 8. **Express ( S ) in Terms of ( a ) and ( c ):** Substitute ( b = a ) and solve for ( S ): [ S = 3a + 3b + c = 3a + 3a + c = 6a + c ] From ( c = 1006 - 2a ), we substitute ( c ) into the equation for ( S ): [ S = 6a + (1006 - 2a) = 4a + 1006 ] 9. **Determine the Range of Possible Values for ( S ):** As ( 0 leq a leq 503 ) (since ( c = 1006 - 2a ) must be non-negative): [ S = 4a + 1006 ] Varies over: [ text{When } a = 0, S = 1006 ] [ text{When } a = 503, S = 4 cdot 503 + 1006 = 2012 + 1006 = 3018 ] From ( 4 cdot 0 + 1006 ) to ( 4 cdot 503 + 1006 ), an arithmetic sequence with common difference 4, providing 504 unique positive integer values. 10. **Conclusion:** The number of different positive integer values that ( S ) can attain is: [ boxed{504} ]