answer:1. **Define the Set ( S )**: Let ( S ) be the set of all numbers ( x ) for which there exist positive integers ( n ) and ( k ) such that ( n ) is odd and ( 0 le x le 1 ). Specifically, ( x = frac{n}{2^k} ). 2. **Show ( A ) Cannot Win for ( x < 0 ) or ( x > 1 )**: - If ( x < 0 ), for any distance ( d ) that ( A ) specifies, ( B ) can always choose the direction "left". This ensures the robot remains to the left of ( (0,0) ) indefinitely. - Similarly, if ( x > 1 ), for any distance ( d ) that ( A ) specifies, ( B ) can always choose the direction "right". This ensures the robot remains to the right of ( (1,0) ) indefinitely. 3. **Show ( A ) Cannot Win if ( x notin S )**: - If ( x notin S ), then ( 2x notin S ). Suppose ( A ) specifies a distance ( d ). The robot can move to either ( (x+d, 0) ) or ( (x-d, 0) ). - Since ( (x+d) + (x-d) = 2x notin S ), at least one of ( x+d ) or ( x-d ) does not belong to ( S ). ( B ) can always choose the direction that results in the robot moving to a position not in ( S ). - Therefore, the robot will never reach ( (0,0) ) or ( (1,0) ). 4. **Show ( A ) Can Win if ( x in S )**: - Let ( x = frac{n}{2^k} ) for some positive integers ( n ) and ( k ) where ( n ) is odd. We will use induction on ( k ) to prove that ( A ) can always win. 5. **Base Case**: - When ( k = 1 ), ( x = frac{n}{2} ) where ( n ) is odd. The only possible value is ( x = frac{1}{2} ). - ( A ) can specify the distance ( frac{1}{2} ). No matter what direction ( B ) chooses, the robot will move to either ( (0,0) ) or ( (1,0) ). 6. **Inductive Step**: - Assume ( A ) has a winning strategy for any ( x = frac{n}{2^k} ) where ( n ) is odd and ( 0 le x le 1 ). - Consider ( x = frac{m}{2^{k+1}} ) for some odd ( m ). ( A ) can specify the distance ( frac{1}{2^{k+1}} ). - The new positions will be ( x + frac{1}{2^{k+1}} ) or ( x - frac{1}{2^{k+1}} ). Both of these positions have denominators that are divisors of ( 2^k ), and by the inductive hypothesis, ( A ) has a winning strategy for these positions. By induction, ( A ) can always win if ( x in S ). (blacksquare) The final answer is all points ( boxed{ P(x,0) } ), where ( x = frac{n}{2^k} ) for some positive integers ( n ) and ( k ) where ( n ) is odd and ( 0 le x le 1 ).

answer:Rotating a point 180 degrees clockwise about the origin results in the transformation of each point (x, y) to (-x, -y). Applying this transformation to point A(-3, 2): [ (-3, 2) rightarrow -(-3), -(2) = (3, -2) ] Conclusion: After rotating point A 180 degrees clockwise about the origin, its image will be at boxed{(3, -2)}.

answer:Since it is known that tanalpha= frac {1}{2}, then sin2alpha= frac {2sinalphacosalpha}{sin^{2}alpha +cos^{2}alpha }= frac {2tanalpha}{1+tan^{2}alpha }= frac {1}{1+ frac {1}{4}}= frac {4}{5}. Therefore, the answer is: boxed{frac {4}{5}}. By using the basic relationship of trigonometric functions of the same angle and the formula for the sine of double angles, we obtain the value of sin2alpha. This question mainly examines the application of the basic relationship of trigonometric functions of the same angle and the formula for the sine of double angles, and it is a basic question.

answer:1. **Calculate the Sale Price After Discount**: The original price of the laptop is 1200 dollars. The discount offered is 30%. The amount of discount in dollars can be calculated as follows: [ text{Discount} = 30% times 1200 = 0.30 times 1200 = 360 text{ dollars} ] Therefore, the sale price after applying the discount is: [ text{Sale Price} = text{Original Price} - text{Discount} = 1200 - 360 = 840 text{ dollars} ] 2. **Calculate the Final Price Including Tax**: A tax of 12% is added to the sale price. The amount of tax in dollars is calculated as: [ text{Tax} = 12% times 840 = 0.12 times 840 = 100.8 text{ dollars} ] Thus, the total selling price of the laptop, including tax, is: [ text{Total Selling Price} = text{Sale Price} + text{Tax} = 840 + 100.8 = 940.8 text{ dollars} ] [ 940.8 text{ dollars} ] Conclusion: The total selling price of the laptop, after accounting for the discount and added tax, is 940.8 dollars. The final answer is The final answer, given the choices, is boxed{940.8} dollars, which corresponds to boxed{text{(C)}}.