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Let \(A\) be the intersection point of the curve \(y=x^2\) and the line \(y=x~(x>0),\) and \(B\) the intersection point of \(y=x^2\) and \(y=-2x~(x<0).\) If \(O=(0, 0)\) is the origin, what is the area of \(\triangle OAB?\)

(A)\(\ \ 2\)

(B)\(\ \ 3\)

(C)\(\ \ 4\)

(D)\(\ \ 5\)

(E)\(\ \ 6\)

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In the above diagram, quadrilateral \(ABCD\) is a square centered at the origin \(O,\) and quadrilateral \(EFGH\) is a rectangle. Suppose that \(A=(-1, 1), E=(2, 0), G=(6, -2),\) and a single line cuts \(\square ABCD\) into halves. If the same line cuts the area of rectangle \(EFGH\) at a ratio of \(1:7,\) the shape with vertex \(F\) being \(1\) and the shape with vertex \(H\) being \(7,\) what is the sum of the \(x\)-coordinates of the points on rectangle \(EFGH\) that intersect the line?

(A)\(\ \ 5.5\)

(B)\(\ \ 6\)

(C)\(\ \ 6.5\)

(D)\(\ \ 7\)

(E)\(\ \ 8\)

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For all real numbers \(m\) and \(n,\) let \(\oplus\) be defined as \[m\oplus n=(1-m)(2+n).\] Then what is the sum of all the values of \(x\) that satisfy \[(x\oplus x)\oplus(x-1)=(x-1)\oplus (x\oplus x)?\]

(A)\(\ \ -2\)

(B)\(\ \ -1\)

(C)\(\ \ 0\)

(D)\(\ \ 1\)

(E)\(\ \ 2\)

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The above is the graph of the quadratic function \(y=ax^2+bx+c.\) Which of the following statements is true?

\(\begin{array}{r r l}
& \text{I.} & a+b-c<0\\

& \text{II.} & b-2a>0\\

& \text{III.} & 4a+b^2-4ac>0\\

\end{array}\)

(A)\(\ \ \) I only

(B)\(\ \ \) II only

(C)\(\ \ \) I and III only

(D)\(\ \ \) II and III only

(E)\(\ \ \) I, II and III

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In the above diagram, quadrilateral \(ABCD\) is a square with side length \(2\) and \(A=(-1, 1),\) and triangle \(EFG\) is an equilateral triangle with side length \(4\) and \(E=(2, 0).\) If a single line cuts into halves both the areas of \(\square ABCD\) and \(\triangle EFG,\) what is the slope of the line?

(A)\(\ \ -\frac{1}{\sqrt{3}}\)

(B)\(\ \ -\frac{\sqrt{3}}{2}\)

(C)\(\ \ \frac{\sqrt{3}-\sqrt{15}}{2}\)

(D)\(\ \ -\frac{2\sqrt{3}}{3}\)

(E)\(\ \ \frac{-\sqrt{3}-\sqrt{15}}{2}\)

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