The West China Mathematical Olympiad (CWMO) is a mathematical competition for middle school students in western provinces of China (including Jiangxi). Organized by the Chinese Mathematical Olympic Committee, it is generally scheduled to be held every year 165438+ 10. The purpose is to encourage middle school students' interest in learning mathematics in the western region. Since the first competition was held in 200 1, so far, the competition has been held for ten times in Xi, Lanzhou, Urumqi, Yinchuan, Chengdu, yingtan, Nanning, Guiyang, Kunming and Taiyuan. 20 1 1 The 11th Mathematical Olympiad in Western China will be held in Yushan No.1 Middle School in Jiangxi.

Edit the competition form in this paragraph.

The contest is divided into two days, 8:00- 12:00. Four questions a day, each question 15, full score 120. According to the results, it will be divided into first, second and third prizes. The top two candidates of each session will be selected for the China National Training Team of the International Mathematical Olympiad in the following year and participate in the selection of IMO national team. Huang, the gold medal winner of the 50th International Mathematical Olympiad in 2009, entered the national training team through the selection of the Western China Mathematical Olympiad.

The 6th Mathematical Olympiad in Western China.

1. Let be the maximum value of a given positive integer. Here you are.

Second, find the smallest positive integer that satisfies the following conditions: For any four different real numbers not less than, there is an arrangement, so that the equation has four different real number roots.

3. As shown in the figure, in the middle, when = 60, the tangent of the circumscribed circle ω passing through this point intersects with the extension line of this point. The sum of the points is on the line segment and the circle Ω respectively, so = 90,. connects and intersects with the points. These three lines are known as * * * points.

(1) Verification: the angular bisector of "Yes";

(2) the value.

4. Let a positive integer not be a complete square number, and prove that for every positive integer, the value of is an irrational number. Here, it represents the largest integer that does not exceed.

Verbs (abbreviation for verb) are all expressed as the sum of squares of two positive integers. Proof: If, then.

6. As shown in the figure, it is the diameter of the circle. The secant passing through this point intersects the circle at two points, which is the diameter of the circumscribed circle, connecting and extending the intersection point of this point. Verification: four * * * circles.

7. Let it be a positive integer not less than 3, and θ is a real number. Prove that if sum is a rational number, then there is a positive integer, so that sum is a rational number.

8. Given a positive integer, find the minimum value, so that for any binary subset of the set, there exists a subset of the set, which satisfies: (1); (2) Yes, both. The number of elements in a finite set is indicated here.