√(x2 - p) + 2 √(x2 - 1) = x have real roots? What are the roots?
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A1. For which real values of p does the equation
√(x2 - p) + 2 √(x2 - 1) = x have real roots? What are the roots? |
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| A2. Given a point A and a segment BC, determine the locus of all points P in space for which ∠APX = 90o for some X on the segment BC. |
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| A3. An n-gon has all angles equal and the lengths of consecutive sides satisfy a1 ≥ a2 ≥ ... ≥ an. Prove that all the sides are equal. |
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| B1. Find all solutions x1, ... , x5 to the five equations xi + xi+2 = y xi+1 for i = 1, ... , 5, where subscripts are reduced by 5 if necessary. |
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| B2. Prove that cos π/7 - cos 2π/7 + cos 3π/7 = 1/2. |
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| B3. Five students A, B, C, D, E were placed 1 to 5 in a contest with no ties. One prediction was that the result would be the order A, B, C, D, E. But no student finished in the position predicted and no two students predicted to finish consecutively did so. For example, the outcome for C and D was not 1, 2 (respectively), or 2, 3, or 3, 4 or 4, 5. Another prediction was the order D, A, E, C, B. Exactly two students finished in the places predicted and two disjoint pairs predicted to finish consecutively did so. Determine the outcome. |
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