Toán học 9

Toán học, Olympic toán toàn quốc - Việt
nam 2002
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.