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1、全国高中数学联赛一、选择题（本题满分36 分，每小题6 分）1正数列满足231221,10,103nnntaaa aan，则100lg()aA、98 B、99 C、100 D、101 2已知 lg x 的小数部分为a，则21lgx的小数部分为A、2a的小数部分B、12a的小数部分C、22a的小数部分D、以上都不正确3过原点O 引抛物线224yxaxa的切线，当a 变化时，两个切点分别在抛物线（）上A、2213,22yxyxB、2235,22yxyxC、22,3yxyxD、223,5yxyx4已知 ABC 为等腰直角三角形，C=90，D、E 为 AB 边上的两个点，且点 D 在 AE 之间，DC
2、E =45，则以AD、DE、ED 为边长构成的三角形的最大角是A、锐角B、钝角C、直角D、不能确定5将正整数从1 开始不间断的写成一行，第2006 个数码是（旁注：这是希望杯的培训题）A、0 B、5 C、7 D、以上都不正确6已知圆锥的顶点V 和底面圆心O 的连线垂直于底面（旁注，这句话实际上是废话），一个过VO 中点 M 的平面与圆O 相切，与圆锥的交线是一个椭圆，若圆O 半径为 1，则椭圆的短轴的长为A、425B、2 33C、62D、以上结果都不对二、（每小题9 分，共 54 分）7设等差数列的首项和公差均为正整数，项数为不小于3 的素数，且各项之和为2006，则这样的数列共有 _个8已知
3、实数x、y 满足55111511541545xxyy，则 xy_（旁注：联赛原题）9正八边形所有对角线在其内部交点的个数为_10若 x、y 为实数，且223xxyy，则22xxyy的最大值和最小值分别为_11一个正方体的8 个顶点可以组成_个非等边三角形12若关于x 的方程212xkx恰有一个实根，则k 的取值范围是_三、论述题（本题满分60 分，每小题20 分）13设有 2006 个互不相同的复数，其中任何两个数的积（包括自乘）是这2006 个数之一，求这2006 个数的和14求的值15已知数列0nan满足00a，对于所有nN，有12 3 011 15nnnnaa aa，求nanknknkk
4、nknknkCnCknCk11122332的通项公式全国高中数学联赛一、填空题（每题10 分，80 分）1、由曲线所围成的几何图形的面积为。2、设实数x，y 满足，则 xy。3、已知,且，则存在整数，使下列等式成立的有个。4、已知平面上两定点A（3，0），B（0，4），P 为曲线上任意一点，过点 P 作 PC x 轴，PDy 轴，垂足分别为C,D，则四边形ABCD 面积 S的最小值为。5、已知均为正实数，且满足，则。6、已知是互不相同的n 个正整数，且满足，则正整数n 的最大值为。7、已知函数，若m，n，则 mn。|23|xyx2011200920112009(1)(1)20104020(1)
5、(1)20100 xxxyyy,a b cR1111abcabck21212121111111()kkkkabcabc2121212121211111kkkkkkabcabc2222111111()kkkkabcabc2222221111kkkkkkabcabc12(0)yxx123,na aaa12312311111,4nnaaaaaaaa123na a aa123,nk k kk33331232024nkkkk22()1xf xx(1)(2)(101)fff1111()()()()23100101ffff文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编
6、码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J
7、5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编
8、码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J
9、5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编
10、码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J
11、5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编
12、码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2S C A B D E 8、已知半径为r 的球和半径为R 的两个相切的球都相切，且它们都与大小为60的二面角的两个半平面相切，则。二、解答题（本大题共4 小题，满分70 分，解答应写出文字说明，证明过程和演算步骤）9、（本题 15 分）已知三棱锥S-ABC 中，SC平面 ABC，AB=BC=CA=，SC2，D,E 分别为 AB，BC 的中点，若点P 在 SE上移动，求 PCD 面积的最小值。10、（本题 20 分）已知定点M（3，0），P，Q
13、 分别是 x 轴及 y 轴上的动点，且使 MPPQ，点 N 在直线 PQ 上，（1）求动点N 的轨迹 C 的方程（2）过点 T(1，0)作直线交轨迹 C 于 A,B 两点，问在x轴上是否存在一点D 使 ABD 为等边三角形，若存在，求出点 D 的坐标，若不存在，请说明理由。11、（本题 20 分）已知数列满足（1）求的通项公式，并证明对任意的x0，（2）求证12、（本题 15 分）平面上有 n个不共线的点，在每个点旁标注数字，如果一条直线通过这些点中的两个或者更多个时，则这些点旁所标注的数Rr4232PNNQlna11412,1,2,.1121nnnaaanana2333(),1,2,.2(2
14、)4nnaxnxx2123321nnaaaan(3)n12,nA AA(1,2,3,)iA inia108642-2-4-6-5510Qg y =0.25y2OTBADPNM文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E
15、7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V
16、1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E
17、7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V
18、1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E
19、7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V
20、1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2字之和为零，证明：所有标注的数字都为零。全国高中数学联赛一填空题（共2 题，每题 10 分，合计 80分）1 设多项式满足：对于任意，都有则的最小值是.2数列满足：已知数列的前项和为，则
21、数列的前项和.3函数的值域是.4过抛物线的焦点,作一条斜率为的直线，若交抛物线于两点，则的面积是.5.若为锐角三角形，满足，则的最大值为.6.若正三棱锥的内切球半径为，则其体积的最小值为.7.将随机填入右图正方形的九个格子中，则其每行三数，每列三数自上而下、自左而右顺次成等差数列的概率.8.将集合的元素分成不相交的三个子集：，其中，且，则集合为：.二解答题（共2 题，合计 70 分）9.（20 分）如图，是圆的一条弦，它将圆分成两部分，分别是两段弧的中点，以点为旋转中心，将弓形顺时针旋转一个角度成弓形，的中点为，的中点为.求证：.()f xxR2(1)(1)24,f xf xxx()f x,n
22、nab1,1,2,kka bknan1nnAnnbnnB21()2xf xx28yxF2ll,A BOABABCsincos()sinAABBtan A11,2,9ABCDp1,2,12MMABC123412341234,Aa a a aBb b b b Cc c c c1c2c3c4ckkkabc1,2,3,4,kCABM、NBAMB1AMB1AAPMNQ2MNPQ文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I
23、7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R
24、5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I
25、7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R
26、5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I
27、7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R
28、5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I
29、7X2Q210.（25 分）给定椭圆以及圆，自椭圆上异于其顶点的任意一点，做的两条切线，切点为，若直线在轴上的截距分别为；证明：.11.（25分）对于个素数组成的集合，将其元素两两搭配成个乘积，得到一个元集，如果与是由此得到的两个元集，其中=，且，就称集合对是由炮制成的一副“对联”.(例如当时，由四元集可炮制成三副“对联”：，).(1).当时，求元素集所能炮制成的“对联”数；（2）对于一般的，求由元素集所能炮制成的“对联”数.指导学生补充证明2222:1,(xyCaabb0)222:O xybPO,M NMN,x y,m n222222abanmb2n122,nMpppnn1234212,nn
30、Aa a a aaa1 234,Bbbb b,212nnbbn122,na aa122,nb bbMAB,A BM2n,a b c d,ab cd,ac bd,ab cd,ad bc,ac bd,ad bc3n6M,a b c d e f2n2nM()T n文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K
31、1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY
32、6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K
33、1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY
34、6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K
35、1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY
36、6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2对金伟华指导学生王慈深参加全国数学联赛的补充说明：指导方式：一对一进行辅导指导方法：以做题讲题为主，指
37、导解题方法指导学时：40 学时（一学时45 分钟）伊春市第二中学20XX 年 1 月 5 日文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7
38、R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1
39、I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7
40、R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1
41、I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7
42、R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1
43、I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2指导学生补充证明对金伟华指导学生丛云鹏参加全国数学联赛的补充说明：指导方式：集中对多个参加联赛学生进行辅导指导方法：以做题讲题为主，指导解题方法指导学时：30 学时（一学时45 分钟）伊春市第二中学20XX 年 1 月 5 日文档编码:CK7V2K1E7R5 HH7I
44、3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文
45、档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I
46、3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文
47、档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I
48、3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文
49、档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I
50、3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2指导学生补充证明对金伟华指导学生初爽参加全国数学联赛的补充说明：指导方式：集中对多个参加联赛学生进行辅导指导方法：以做题讲题为主，指导解题方法指导学时：30 学时（一学时45 分钟）伊春市第二中学20XX 年 1 月 5 日文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V2K1E7R5 HH7I3J5X6B5 ZY6V1I7X2Q2文档编码:CK7V