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The 40th Vietnam Mathematical Olympiad
2002年第40届越南数学奥林匹克

A1. Solve the following equation: √(4 - 3√(10 - 3x)) = x - 2 .

A2. ABC is an isosceles triangle with AB = AC. O is a variable point on the line BC such that the circle center O radius OA does not have the lines AB or AC as tangents. The lines AB, AC meet the circle again at M, N respectively. Find the locus of the orthocenter of the triangle AMN .

A3. m < 2001 and n < 2002 are fixed positive integers. A set of distinct real numbers are arranged in an array with 2001 rows and 2002 columns. A number in the array is bad if it is smaller than at least m numbers in the same column and at least n numbers in the same row. What is the smallest possible number of bad numbers in the array ?

B1. If all the roots of the polynomial x3 + a x2 + bx + c are real, show that 12ab + 27c ≤ 6a3 + 10(a2 - 2b)3/2. When does equality hold ?

B2. Find all positive integers n for which the equation a + b + c + d = n√(abcd) has a solution in positive integers .

B3. n is a positive integer. Show that the equation 1/(x - 1) + 1/(22x - 1) + ... + 1/(n2x - 1) = 1/2 has a unique solution xn > 1. Show that as n tends to infinity, xn tends to 4 .

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