1978年全国高中数学竞赛题（National high school mathematics competition in 1978）.docVIP

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1978年全国高中数学竞赛题（National high school mathematics competition in 1978）.doc

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1978年全国高中数学竞赛题（National high school mathematics competition in 1978）

1978年全国高中数学竞赛题（National high school mathematics competition in 1978） In 1978 the national high school mathematics competition problems A test The 1. known y=log121x+3, when asked why the x value, y0 (I); (II) y0? 2. known tanx=22 (180 degrees x270), for cos2x, the value of cosx2. 3. is the center of the ellipse is the origin, it is on the X axis with a focus on both ends of the short axis line perpendicular to each other, the closer the distance endpoint and the focus and the long axis is 10 - 5 for elliptic equations. 4. known for a 2x29x+8=0 equation, quadratic equation, so it is a root of the original equation and the last two, another root is the square of the difference between the two original equations. 5. the radius is four fold 1 balls into two layers: the bottom layer on the table three, the upper one, 22 for the highest point of tangency, the upper ball from the height of the desktop. As shown in figure 6., the midpoint set of line AB M, leading the line CD from another C line on the AB to the side of the line AB, the midpoint segment of CD N, BD P MN point, Q point, line bisection line AC. PQ confirmation: 7. proof: when n and K are positive integers, and given n2, k2, n (n 1) K 1 can be written as N and even continuous. 8. show: three the cosine of the angle of the triangle and less than half Zhou Changzhi triangle vertex on the unit circle. The 9. known line l1:y=4x and point P (6, 4), for a little Q on line L1, the PQ line and L1 and X axis in the first quadrant of enclosing triangle area is minimum. 10. equations x+y+z=0, X3+y3+z3=18 integer solutions. Two questions The 1. group of two quadrilateral boundary extension respectively intersect, and connect with the quadrilateral intersection of a diagonal line parallel to prove: the other diagonal extension line bisection of line edge intersections connected. 2. - factorization: x12+x9+x6+x3+1. The proof: for any angle, there are 5+8cos +4cos2 +cos3 theta theta theta = 0. The 3. R plane to A (4, 1), B (