toán học olympic toán toàn quốc việt nam 2002 toán học olympic toán toàn quốc việt nam 2002 bài từ tủ sách khoa học vlos currently 5 005 bài viết xuất sắc 5 05 1 vote jump to navigation search

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<b>Tốn học, Olympic tốn tồn quốc - Việt </b>

<b>nam 2002</b>

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.

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