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Official Solution:
Is \(xy < 0\)?
Note that the question essentially asks whether \(x\) and \(y\) have different signs.
(1) \(x^2 = y^2\)
This statement implies that \(|x| = |y|\), which means that \(x\) and \(y\) are equidistant from 0. This information is not sufficient to determine whether \(x\) and \(y\) have different signs. For example, consider \(x = 1\) and \(y = 1\), and \(x = 1\) and \(y = -1\).
(2) \(\frac{1}{x + y} < 1\)
This statement is also not sufficient to determine whether \(x\) and \(y\) have different signs. For example, consider \(x = -1\) and \(y = -1\), and \(x = 1\) and \(y = -2\).
(1)+(2) From (1), we deduce that either \(x = y\) or \(x = -y\). However, if \(x = -y\), then \(x + y = 0\), which contradicts statement (2) since \(\frac{1}{x + y} < 1\) and the expression \(\frac{1}{x + y}\) would be undefined when \(x + y = 0\), not less than 1. Therefore, we must have \(x = y\), which implies that \(x\) and \(y\) have the same sign. This leads to a NO answer to the question "Is \(xy < 0\)?". Sufficient.
Answer: C
Is \(xy < 0\)?
Note that the question essentially asks whether \(x\) and \(y\) have different signs.
(1) \(x^2 = y^2\)
This statement implies that \(|x| = |y|\), which means that \(x\) and \(y\) are equidistant from 0. This information is not sufficient to determine whether \(x\) and \(y\) have different signs. For example, consider \(x = 1\) and \(y = 1\), and \(x = 1\) and \(y = -1\).
(2) \(\frac{1}{x + y} < 1\)
This statement is also not sufficient to determine whether \(x\) and \(y\) have different signs. For example, consider \(x = -1\) and \(y = -1\), and \(x = 1\) and \(y = -2\).
(1)+(2) From (1), we deduce that either \(x = y\) or \(x = -y\). However, if \(x = -y\), then \(x + y = 0\), which contradicts statement (2) since \(\frac{1}{x + y} < 1\) and the expression \(\frac{1}{x + y}\) would be undefined when \(x + y = 0\), not less than 1. Therefore, we must have \(x = y\), which implies that \(x\) and \(y\) have the same sign. This leads to a NO answer to the question "Is \(xy < 0\)?". Sufficient.
Answer: C
