![](https://wkrtcs.bdimg.com/rtcs/image?w=34.666666666667&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"34.666666666667","h":"49.333333333333","dataType":"png","c":"word/media/image1.png","imgOriW":26,"imgOriH":37})
2018年福建省高一数学竞赛试题
一、选择题：本大题共6个小题,每小题6分,共36分.在每小题给出的四个选项中，只有一项是符合题目要求的.
1.已知集合![](https://wkrtcs.bdimg.com/rtcs/image?w=113&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"113","h":"27","dataType":"gif","c":"word/media/image2.png"})
，，则（ ）
A．![](https://wkrtcs.bdimg.com/rtcs/image?w=40&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"40","h":"20","dataType":"gif","c":"word/media/image5.png"})
B． C． D．
2.若直线![](https://wkrtcs.bdimg.com/rtcs/image?w=9&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"9","h":"17","dataType":"gif","c":"word/media/image9.png"})
与两直线：，：分别交于，两点，且线段中点为，则直线的斜率为（ ）
A．![](https://wkrtcs.bdimg.com/rtcs/image?w=20&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"20","h":"16","dataType":"gif","c":"word/media/image18.png"})
B． C． D．
3.如图，在正方体![](https://wkrtcs.bdimg.com/rtcs/image?w=105&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"105","h":"21","dataType":"gif","c":"word/media/image22.png"})
中，、分别为棱、的中点，为正方形的中心.为平面与平面与平面的交线，则直线与正方体底面所成角的大小为（ ）
![](https://wkrtcs.bdimg.com/rtcs/image?w=121&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"121","h":"126","dataType":"png","c":"word/media/image32.png","imgOriW":216,"imgOriH":224})
A．![](https://wkrtcs.bdimg.com/rtcs/image?w=24&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"24","h":"17","dataType":"gif","c":"word/media/image33.png"})
B． C． D．
4.如图，在三棱锥![](https://wkrtcs.bdimg.com/rtcs/image?w=55&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"55","h":"17","dataType":"gif","c":"word/media/image37.png"})
中，，且侧面底面，则三棱锥外接球的表面积为（ ）
![](https://wkrtcs.bdimg.com/rtcs/image?w=101&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"101","h":"137","dataType":"png","c":"word/media/image41.png","imgOriW":180,"imgOriH":244})
A．![](https://wkrtcs.bdimg.com/rtcs/image?w=28&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"28","h":"17","dataType":"gif","c":"word/media/image42.png"})
B． C. D．
5.已知定义在![](https://wkrtcs.bdimg.com/rtcs/image?w=15&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"15","h":"16","dataType":"gif","c":"word/media/image46.png"})
上的函数满足：且，，则方程在区间上的所有实根之和为（ ）
A．![](https://wkrtcs.bdimg.com/rtcs/image?w=17&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"17","h":"16","dataType":"gif","c":"word/media/image53.png"})
B． C. D．
6.已知点![](https://wkrtcs.bdimg.com/rtcs/image?w=59&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"59","h":"20","dataType":"gif","c":"word/media/image57.png"})
，，，直线（）交线段于点，交线段于点.若的面积为，则的取值范围为（ ）
A．![](https://wkrtcs.bdimg.com/rtcs/image?w=68&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"68","h":"24","dataType":"gif","c":"word/media/image67.png"})
B． C. D．
二、填空题（每题6分，满分36分，将答案填在答题纸上）
7.函数![](https://wkrtcs.bdimg.com/rtcs/image?w=199&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"199","h":"41","dataType":"gif","c":"word/media/image71.png"})
的最小值为 ．
8.如图，在四棱锥![](https://wkrtcs.bdimg.com/rtcs/image?w=65&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"65","h":"17","dataType":"gif","c":"word/media/image72.png"})
中，平面，底面为正方形，.、分别为、的中点，则二面角的正切值为 ．
![](https://wkrtcs.bdimg.com/rtcs/image?w=104&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"104","h":"116","dataType":"png","c":"word/media/image78.png","imgOriW":167,"imgOriH":187})
9.若函数![](https://wkrtcs.bdimg.com/rtcs/image?w=140&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"140","h":"23","dataType":"gif","c":"word/media/image79.png"})
在区间（）上的值域为，则实数的取值范围为 ．
10.已知集合![](https://wkrtcs.bdimg.com/rtcs/image?w=124&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"124","h":"23","dataType":"gif","c":"word/media/image84.png"})
，集合，则集合中元素的个数为 ．
11.使![](https://wkrtcs.bdimg.com/rtcs/image?w=64&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"64","h":"41","dataType":"gif","c":"word/media/image86.png"})
为有理数的所有正整数的和为 ．
12.给出下列![](https://wkrtcs.bdimg.com/rtcs/image?w=17&md5sum=b260631f21e4db36ef196e5387391adf&sign=a649787cb8&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=196&ipr={"t":"img","w":"17","h":"17","dataType":"gif","c":"word/media/image88.png"})
个数：，，，，，，，，，，其中，，为整数，且.若对每个正整数，都可以表示成上述个数中某些数的和（可以是个数的和，也可以是个数的和，每个数至多出现次），则的最小值为 ．
三、解答题 （本大题共5小题，共78分.解答应写出文字说明、证明过程或演算步骤.）
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