07 Shanghai High School Mathematics Competition (Xinzhi Cup) Examination Paper
(Sunday, March 25th, 2007, 8: 30 am-10: 30 am)
Explain that you must not use a calculator to solve this test paper.
Fill-in-the-blank question (the full score of this question is 60, the first four questions are 7 points for each small question, and the last four questions are 8 points for each small question)
The real number solution of the equation _ _ _ _ _ _ _.
There is a line segment with a length, and its endpoints slide on four sides of a square with a side length of. When it slides around the four sides of a square, the length of the trajectory formed by the points is _ _ _ _ _ _ _ _ _ _ _.
If the complex sequence satisfies (,which is an imaginary unit), the sum of the preceding paragraphs is _ _ _ _ _ _ _ _.
The dihedral angle of a known size is a certain point in the dihedral angle, and the distances to the sum of the half planes are sum, which is the moving point in the half plane, so the minimum perimeter is _ _ _ _ _ _ _ _ _ _.
The corresponding laws between points in the plane rectangular coordinate system are known. If a curve corresponds to an arc of an ellipse under the corresponding rules, then the equation of this curve is _ _ _ _ _ _ _ _ _ _ _.
Known, calculation: _ _ _ _ _ _ _ _ _ _.
A given series satisfies,,, then the general formula of the series is _ _ _ _ _ _ _ _ _.
It is known that the circle has a secant at the point passing through the axis, so the value range of the abscissa of this point is _ _ _ _ _ _ _ _ _ _ _ _.
answer the question
(The full mark of this question is 14) For any positive integer, use the number of positive integer pairs satisfying the indefinite equation. For example, if there are three positive integer pairs, all positive integers that satisfy are found.
The full mark of this question is 14. It is known that the equation has three positive real roots, so find the minimum value.
(Full score of this question 16) As we all know, a parabola is an over-focused string. If the angle with the axis is, find.
(Full score of this question 16) Find the smallest positive integer satisfying the following conditions. Take any point on the circumference,,, and at least one angle will not exceed.
Answers to the examination papers of 2007 Shanghai Senior High School Mathematics Competition
1.(2,8, 18) 2.3.– 1003+2 i4。
5.6.
7.8
9. Or or or or or or
10
1 1.
12 9 1