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A1. Find all x in the interval [0, 2π] which satisfy:
2 cos x ≤ |√(1 + sin 2x) - √(1 - sin 2x)| ≤ √2. |
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A2. The coefficients aij of the following equations
a11x1 + a12 x2+ a13 x3 = 0 satisfy the following: (a) a11, a22, a33 are positive, (b) other aij are negative, (c) the sum of the coefficients in each equation is positive. Prove that the only solution is x1 = x2 = x3 = 0. |
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| A3. The tetrahedron ABCD is divided into two parts by a plane parallel to AB and CD. The distance of the plane from AB is k times its distance from CD. Find the ratio of the volumes of the two parts. |
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| B1. Find all sets of four real numbers such that the sum of any one and the product of the other three is 2. |
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| B2. The triangle OAB has ∠O acute. M is an arbitrary point on AB. P and Q are the feet of the perpendiculars from M to OA and OB respectively. What is the locus of H, the orthocenter of the triangle OPQ (the point where its altitudes meet)? What is the locus if M is allowed to vary over the interior of OAB? |
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| B3. Given n > 2 points in the plane, prove that at most n pairs of points are the maximum distance apart (of any two points in the set). |
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