The answers to the preliminaries of the 2009 National Senior High School Mathematics League in Jiangxi Province can be downloaded directly for free/a/shuxuejingsai/2009/1123/87.html1. Fill in the blanks (* * * questions, each question 10 points, score) 1. Then the array solution: Note that for an integer, if the last digit of is, then the last digit of must be or, easy to know, (), so if the condition is to be met, it can only be, because, so, .2. If the focus and vertex of an ellipse are the vertices and focus of a hyperbola, then the equation of an elliptic circle is: Solution: The two vertices of a hyperbola are two. Solution: make, then, by, get, because it is a real number, then the discriminant, get .4, the dihedral angle between the plane and the plane in the tetrahedron, then the distance from the point to the plane is. Solution: Make a plane, the vertical foot is even, and the reason is that the three vertical lines are reversed, so, so, and because it is a square, so, it is a regular triangle. 5. After removing all multiples of the sum of multiples from the set, the number of remaining elements is. Solution: In a set, there are multiples, multiples and multiples, so the number of remaining elements is. The range of the function is. Solution: make, and then, in this way, the two sides get equal symbols respectively. Solution: Original formula. (Note that. ), nine consecutive positive integers are arranged in a row from small to large. If it is a square number and a cubic number, the minimum value of the sum of these nine positive integers is. Solution: Let these nine numbers be, then,,, and then, get the total score of, (20 points) for a point () on the given axis. For the moving point on the curve, try to find the minimum value of the distance between two points (expressed by). Solution: As shown in the figure, it is easy to find the coordinates of the points on the curve: if, immediately, the curve equation is … when, the curve equation is ②, for case ①, immediately, obviously, when it is at the vertex, the distance gets the minimum; ....................................................................................................................................................... ..........................., if, then, The minimum value is .....................................................................................................................................................................'s proof: three-point * * * line. Proof: As shown in the figure, let it be three disjoint chords, where,, and let it be three sides of a point. According to Menelius' inverse theorem, As long as proved (1), Conclusion It is proved that the sum of the items in the positive integer sequence. ....................................................................................................................................... A point in .............. represents a group. If this group has a common term, then there is an edge between the corresponding points, thus a bipartite graph is obtained. This bipartite graph has exactly one edge and one vertex ................................................................................................................................., and in a group, Every number of should also appear in a group, (otherwise there will be edges connected with vertices outside the branch, which will lead to contradictions), so the sum of numbers in a group should be equal to the sum of numbers in a group, that is, there is, from where, so, contradiction! Therefore, it is a connected graph. So the graph has at least one edge, namely; Items of value; In addition, the other seven are divided into seven pairs (including four pairs, two pairs, one pair and another item), so each category group can be composed of one, one or one pair; In this way, * * * gets a class group, and the sum of the numbers in each group is; In order to get a set of sums, we can make each group, and the remaining numbers can be spelled into eight groups: four in the sum group, two in the sum group, one in the sum group and one in the sum group. Therefore, the minimum value is ... 25 points.

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