from: category_eng |
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Define to be for all real numbers and . Which of the following statements is not true? ' |
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Let , , , and be real numbers with , , and . What is the sum of all possible values of ? ' |
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Examining statement C: when , but statement C says that it does for all . Therefore the statement that is not true is |
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is the distance between and ; is the distance between and . Therefore, the given equation says is equidistant from and , so . Alternatively, we can solve by casework (a method which should work for any similar problem involving absolute values of real numbers). If , then and , so we must solve , which has no solutions. Similarly, if , then and , so we must solve , which also has no solutions. Finally, if , then and , so we must solve , which has the unique solution . |
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Solution 1Substitution gives . This gives . There are possibilities for the value of : Therefore, the only possible values of are 9, 5, 3, and 1. Their sum is . Solution 2If we add the same constant to all of , , , and , we will not change any of the differences. Hence we can assume that . If we multiply all four numbers by , we will not change any of the differences. Hence we can WLOG assume that . Hence , and the sum of possible values is . |