Problem 1. A three-man jury has two members each of whom independently has probability p of making the correct decision and a third member who flips a coin for each decision (majority rules). A one-man jury has the probability p of making the correct decision. Which jury has the better probability of making the correct decision (the 3-member jury or the one-member jury)?
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Problem 2. If a + b + c = 7 and (1)/(a + b) + (1)/(b + c) + (1)/(c + a) = (7)/(10), what is (a)/(b + c) + (b)/(c + a) + (c)/(a + b)?
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Problem 3. Solve the equation xlog29 + x2(3log2x) = xlog215.
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Problem 4. Give the expression of a function with the (maximum possible) domain [0,4] and the range [2,5].
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Problem 5. Let ABCD be a square. The midpoint O of the segment AD is the center of the circle with the diameter AD. If E ∈ AB and EC is tangent to the circle O, prove that BEC < 60.
figure Problem_5/fig_Problem_6.jpg
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Problem 6. For each parabola y = x2 + px + q meeting the coordinate axes in three distinct points, a circle through these points is drawn. Show that all of the circles pass through a single point which is (0,1).
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Problem 7. Consider the curves x2 + y2 = 1 and 2x2 + 2xy + y2–2x–2y = 0. These curves intersect at two points, one of which is (1, 0). Find the other one.
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Problem 8. In the square ABCD we choose the point E such that each of the angles EBC and ECB is 15°. Prove that ADE is equilateral.
figure Problem_8/fig_Problem_8_1..jpg
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