Toán học, Olympic toán toàn quốc - Việt nam 2002 Bài từ Tủ sách Khoa học VLOS.. O is a variable point on the line BC such that the circle center O radius OA does not have the lines AB or
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Bài từ Tủ sách Khoa học VLOS
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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 d" 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