Algebra
# Functions

For positive integer \(k\), let \(\bullet k\bullet = k^{2}-(k-1)^{2}\). What is the value of \(\bullet 7\bullet\)?

(A) \(\ \ -36\)

(B) \(\ \ 7\)

(C) \(\ \ 13\)

(D) \(\ \ 49\)

(E) \(\ \ 85\)

For positive integers \(m\) and \(n\), let \(m \nabla n\) be defined as \(m \nabla n = (\frac{m}{n})^{-1} - (\frac{n}{m})^{-1}\). What is the value of \(2 \nabla 15\)?

(A) \(\ \ -\frac{15}{2}\)

(B) \(\ \ -\frac{221}{30}\)

(C) \(\ \ \frac{221}{30}\)

(D) \(\ \ \frac{2}{15}\)

(E) \(\ \ \frac{229}{30}\)

For positive integers \(m\), \(k\), and \(n\), let \(m\breve{k}n\) be defined as \(m\breve{k}n = k\frac{m}{n}\), where \(k\frac{m}{n}\) is a mixed fraction. What is the value of \(6\breve{4}1 + 1\breve{4}6\)?

(A) \(\ \ 0\)

(B) \(\ \ 5\)

(C) \(\ \ \frac{85}{6}\)

(D) \(\ \ 700\)

(E) \(\ \ 787\)

For positive integers, \(x\), \(y\), and \(z\), let \(\widehat{x,y,z}\) be defined by \(\widehat{x,y,z} = zx - yz - xy\). What is the value of \(m\) if \( \widehat{10,m,10} = \widehat{7,3,1}?\)

(A) \(\ \ -\frac{631}{20}\)

(B) \(\ \ -\frac{117}{20}\)

(C) \(\ \ 0\)

(D) \(\ \ \frac{21}{100}\)

(E) \(\ \ \frac{117}{20}\)

For a positive integer \(n\), let \(\fbox{n} = 1-2+3-4 \ldots n\). For example, \(\fbox{5}=1-2+3-4+5=3.\) Which of the following is equal to \( \fbox{6}-\fbox{48}\)?

(A) \(\ \ -1155\)

(B) \(\ \ -27\)

(C) \(\ \ -21\)

(D) \(\ \ -\fbox{6}\)

(E) \(\ \ 21\)

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