2016年全国高中数学联赛四川赛区预赛试题
适用年级:高三试卷类型:竞赛试题总数:16浏览次数:88上传日期:2018/12/5
一、单选题
1 . 在△ABC中,∠A、∠B、∠C的对边长分别为a、b、c.命题

,且b+c=2a;命题

为正三角形.则命题P是命题q的()条件.
A. 充分必要
B. 充分但不必要
C. 必要但不充分
D. 既不充分又不必要
难度系数:0.65使用:14次题型:单选题更新:2018/12/5
难度系数:0.65使用:15次题型:单选题更新:2018/12/5
3 . 已知函数

,当

时,记

的最小值为m.则m的最大值为().
A.-2 | B.0 | C. | D.l |
难度系数:0.65使用:14次题型:单选题更新:2018/12/5
4 . 对任意正整数n与

,

表示不超过

(

表示不超过实数x的最大整数)且与n互素的正整数的个数.则

().
难度系数:0.4使用:12次题型:单选题更新:2018/12/5
5 . 设数列

满足

,

.若

为

的前n项和,则

的个位数字为().
难度系数:0.65使用:20次题型:单选题更新:2018/12/5
6 . 已知

、

为椭圆与双曲线的公共焦点,P为它们的一个公共点,且

.则该椭圆与双曲线的离心率之积的最小值为().
难度系数:0.65使用:25次题型:单选题更新:2018/12/5
二、填空题
7 . 在

的展开式中,

的系数是__________.
难度系数:0.4使用:616次题型:填空题更新:2018/2/27
8 . 若实数

、

、

构成以2为公比的等比数列,sinα、sinβ、sinγ构成等比数列,则cosa=_____.
难度系数:0.65使用:11次题型:填空题更新:2018/12/5
9 . 已知正四棱锥S-ABCD的侧棱长为4,

,在侧棱SB、SC、SD分别取点E、F、G则空间四边形AEFG周长的最小值为________.
难度系数:0.65使用:12次题型:填空题更新:2018/12/5
10 . 已知△ABC的外心为O,且

,则

________.
难度系数:0.65使用:18次题型:填空题更新:2018/12/5
11 . 设实数x、y、z、w满足

.则

的最大值为________.
难度系数:0.4使用:15次题型:填空题更新:2018/12/5
12 . 对于任何集合S,用

表示集合S中的元素个数,用n(S)表示集合S的子集个数.若A、B、C为三个有限集,且满足(1)

;(2)

.则

的最大值为_______.
难度系数:0.15使用:19次题型:填空题更新:2018/12/5
三、解答题
13 . 设等比数列

的前n项和为

(r为常数).记

.
(1)求数列

的前n项和

;
(2)若对于任意的正整数n,均有

,求实数k的最大值.
难度系数:0.65使用:12次题型:解答题更新:2018/12/5
14 . 已知a、b、c为正实数.证明:

.
难度系数:0.4使用:11次题型:解答题更新:2018/12/5
15 . 已知拋物线

过定点C(l,2),在抛物线上任取不同于点C的一点A,直线AC与直线y=x+3交于点P,过点P作x轴的平行线,与抛物线交于点B.
(1)证明:直线AB过定点;
(2)求△ABC面积的最小值.
难度系数:0.65使用:20次题型:解答题更新:2018/12/5
16 . 已知a为实数,函数

.请讨论函数

单调性.
难度系数:0.4使用:12次题型:解答题更新:2018/12/5
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src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image005.png\" style=\"vertical-align:middle;\" />,则<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image006.png\" style=\"vertical-align:middle;\" />()<table name=\"optionsTable\" cellpadding=\"0\" cellspacing=\"0\" width=\"100%\"><tr><td width=\"25%\">A.-1</td><td width=\"25%\">B.<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image007.png\" style=\"vertical-align:middle;\" /></td><td width=\"25%\">C.<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image008.png\" style=\"vertical-align:middle;\" /></td><td 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id=\"80254\">选修2-3</category><category id=\"80255\">第一章 计数原理</category><category id=\"80261\">1.3 二项式定理</category></item><item><category id=\"79903\">人教A版</category><category id=\"80254\">选修2-3</category><category id=\"80255\">第一章 计数原理</category><category id=\"80261\">1.3 二项式定理</category><category id=\"80265\">1.3.1 二项式定理</category></item><item><category id=\"79903\">人教A版</category><category id=\"80254\">选修2-3</category><category id=\"80255\">第一章 计数原理</category><category id=\"80261\">1.3 二项式定理</category><category id=\"80266\">1.3.2 “杨辉三角”与二项式系数的性质</category></item><item><category id=\"80512\">人教B版</category><category id=\"80844\">选修2-3</category><category id=\"80845\">第一章 计数原理</category><category id=\"80851\">1.3 二项式定理</category></item><item><category id=\"80512\">人教B版</category><category id=\"80844\">选修2-3</category><category id=\"80845\">第一章 计数原理</category><category id=\"80851\">1.3 二项式定理</category><category id=\"80855\">1.3.1 二项式定理</category></item><item><category id=\"81004\">苏教版</category><category id=\"81303\">选修2-3</category><category id=\"81304\">第1章 计数原理</category><category id=\"81312\">1.5 二项式定理</category></item><item><category id=\"81004\">苏教版</category><category id=\"81303\">选修2-3</category><category id=\"81304\">第1章 计数原理</category><category id=\"81312\">1.5 二项式定理</category><category id=\"81314\">1.5.1 二项式定理</category></item><item><category id=\"81004\">苏教版</category><category id=\"81303\">选修2-3</category><category id=\"81304\">第1章 计数原理</category><category id=\"81312\">1.5 二项式定理</category><category id=\"81315\">1.5.2 二项式系数的性质及应用</category></item><item><category id=\"81004\">苏教版</category><category id=\"81637\">高中三年级</category><category id=\"81640\">第16章 排列组合和二项式定理</category><category id=\"81665\">16.5 二项式定理</category></item><item><category id=\"81720\">湘教版</category><category id=\"82019\">选修2-3</category><category id=\"82020\">第7章 计数原理</category><category id=\"82026\">7.4 二项式定理</category></item><item><category id=\"82158\">大纲版</category><category id=\"82319\">高二下册</category><category id=\"82321\">第十章 排列、组合和二项式定理</category><category id=\"82341\">10.4 二项式定理</category></item>","ChildNum":1,"Time":"2018-02-27T11:53:01","QuesBody":"<div>【题文】在<img src=\"Upload/2018-07/20/285f463e-c6e6-4955-be34-754e93c5e372/paper.files/image001.png\" style=\"vertical-align:middle;\" />的展开式中,<img src=\"Upload/2018-07/20/285f463e-c6e6-4955-be34-754e93c5e372/paper.files/image002.png\" style=\"vertical-align:middle;\" />的系数是__________.</div>","QuesAnswer":"","QuesParse":"","UseSum":616,"AvgScore":5.0,"ParseAnswerKey":"a0ac848a92224c27e131d35568ac6021","QuesAttribute":0},{"ID":7264419,"Guid":"b312971d-a0ce-43a3-a1f4-5689719ecc81","Title":"2016年全国高中数学联赛四川赛区预赛试题","From":"","Knowledge":"","QuesType":{"ID":2702,"Name":"填空题","IsSelectType":false,"OrderIndex":3,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132858\">三角函数</category><category id=\"132892\">三角恒等式、不等式、最值</category></item>","ChildNum":1,"Time":"2018-12-05T19:50:38","QuesBody":"<div>【题文】若实数<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image073.png\" style=\"vertical-align:middle;\" />、<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image074.png\" style=\"vertical-align:middle;\" />、<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image075.png\" style=\"vertical-align:middle;\" />构成以2为公比的等比数列,sinα、sinβ、sinγ构成等比数列,则cosa=_____.</div>","QuesAnswer":"","QuesParse":"","UseSum":11,"AvgScore":5.0,"ParseAnswerKey":"343b89ab6b02fd1c65957ec7c2de63e4","QuesAttribute":2},{"ID":7264420,"Guid":"cc3ea2de-6cd0-48b2-a8de-47e94dd83af9","Title":"2016年全国高中数学联赛四川赛区预赛试题","From":"","Knowledge":"","QuesType":{"ID":2702,"Name":"填空题","IsSelectType":false,"OrderIndex":3,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132863\">立体几何</category><category id=\"132940\">表面展开图</category></item>","ChildNum":1,"Time":"2018-12-05T19:50:38","QuesBody":"<div>【题文】已知正四棱锥S-ABCD的侧棱长为4,<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image085.png\" style=\"vertical-align:middle;\" />,在侧棱SB、SC、SD分别取点E、F、G则空间四边形AEFG周长的最小值为________.</div>","QuesAnswer":"","QuesParse":"","UseSum":12,"AvgScore":5.0,"ParseAnswerKey":"84f2bcc87959bc0b93c599a98e770d72","QuesAttribute":2},{"ID":7264421,"Guid":"39a1625e-8b31-4870-9270-cfadefcf6276","Title":"2016年全国高中数学联赛四川赛区预赛试题","From":"","Knowledge":"","QuesType":{"ID":2702,"Name":"填空题","IsSelectType":false,"OrderIndex":3,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132859\">向量</category><category id=\"132896\">向量的运算</category></item><item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132859\">向量</category><category id=\"132897\">向量的坐标表示,数量积</category></item>","ChildNum":1,"Time":"2018-12-05T19:50:39","QuesBody":"<div>【题文】已知△ABC的外心为O,且<img src=\"Upload/2019-01/14/808ef1db-a7d7-4bac-9f86-3abf98214d6a/paper.files/image001.png\" style=\"vertical-align:middle;\" />,则<img src=\"Upload/2019-01/14/808ef1db-a7d7-4bac-9f86-3abf98214d6a/paper.files/image002.png\" style=\"vertical-align:middle;\" />________.</div>","QuesAnswer":"","QuesParse":"","UseSum":18,"AvgScore":5.0,"ParseAnswerKey":"af9caf904a93417e7cd414d6cff11643","QuesAttribute":2},{"ID":7264422,"Guid":"31f60875-b6cc-4c0a-94dd-c9e6cfc0740e","Title":"2016年全国高中数学联赛四川赛区预赛试题","From":"","Knowledge":"","QuesType":{"ID":2702,"Name":"填空题","IsSelectType":false,"OrderIndex":3,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":4,"DifValue":0.4,"Name":"较难"},"Categories":"<item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132861\">不等式</category><category id=\"132903\">重要不等式</category><category id=\"132906\">均值不等式</category></item>","ChildNum":1,"Time":"2018-12-05T19:50:39","QuesBody":"<div>【题文】设实数x、y、z、w满足<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image100.png\" style=\"vertical-align:middle;\" />.则<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image101.png\" style=\"vertical-align:middle;\" />的最大值为________.</div>","QuesAnswer":"","QuesParse":"","UseSum":15,"AvgScore":5.0,"ParseAnswerKey":"59ff2e99b7bf7520af9fae9ee93f93b7","QuesAttribute":2},{"ID":7264423,"Guid":"2cbd4a46-b292-4b54-ab46-99581a4c7f2e","Title":"2016年全国高中数学联赛四川赛区预赛试题","From":"","Knowledge":"","QuesType":{"ID":2702,"Name":"填空题","IsSelectType":false,"OrderIndex":3,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":5,"DifValue":0.15,"Name":"困难"},"Categories":"<item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132455\">集合</category><category id=\"132875\">子集,子集族</category></item>","ChildNum":1,"Time":"2018-12-05T19:50:40","QuesBody":"<div>【题文】对于任何集合S,用<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image110.png\" style=\"vertical-align:middle;\" />表示集合S中的元素个数,用n(S)表示集合S的子集个数.若A、B、C为三个有限集,且满足(1)<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image111.png\" style=\"vertical-align:middle;\" />;(2)<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image112.png\" style=\"vertical-align:middle;\" />.则<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image113.png\" style=\"vertical-align:middle;\" />的最大值为_______.</div>","QuesAnswer":"","QuesParse":"","UseSum":19,"AvgScore":5.0,"ParseAnswerKey":"5c54b6dd87eda0768627cdb3dcdd39a2","QuesAttribute":2},{"ID":7264424,"Guid":"cb225661-b9ef-4c9a-8629-2dea051b0ad7","Title":"2016年全国高中数学联赛四川赛区预赛试题","From":"","Knowledge":"","QuesType":{"ID":2703,"Name":"解答题","IsSelectType":false,"OrderIndex":4,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132860\">数列</category><category id=\"132899\">数列求和</category></item><item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132861\">不等式</category><category id=\"132904\">证明不等式的常用方法</category><category id=\"132926\">构造法</category></item>","ChildNum":1,"Time":"2018-12-05T19:50:40","QuesBody":"<div>【题文】设等比数列<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image027.png\" style=\"vertical-align:middle;\" />的前n项和为<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image126.png\" style=\"vertical-align:middle;\" />(r为常数).记<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image127.png\" style=\"vertical-align:middle;\" />.<br />(1)求数列<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image128.png\" style=\"vertical-align:middle;\" />的前n项和<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image129.png\" style=\"vertical-align:middle;\" />;<br />(2)若对于任意的正整数n,均有<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image130.png\" style=\"vertical-align:middle;\" />,求实数k的最大值.</div>","QuesAnswer":"","QuesParse":"","UseSum":12,"AvgScore":5.0,"ParseAnswerKey":"45b24c8188f2a396b6164fd0a87232fb","QuesAttribute":2},{"ID":7264425,"Guid":"00597ed9-fbaf-43d0-bf33-0f8ee658e2e5","Title":"2016年全国高中数学联赛四川赛区预赛试题","From":"","Knowledge":"","QuesType":{"ID":2703,"Name":"解答题","IsSelectType":false,"OrderIndex":4,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":4,"DifValue":0.4,"Name":"较难"},"Categories":"<item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132861\">不等式</category><category id=\"132903\">重要不等式</category><category id=\"132906\">均值不等式</category></item><item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132861\">不等式</category><category id=\"132904\">证明不等式的常用方法</category><category id=\"132924\">其他</category></item>","ChildNum":1,"Time":"2018-12-05T19:50:41","QuesBody":"<div>【题文】已知a、b、c为正实数.证明:<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image151.png\" style=\"vertical-align:middle;\" />.</div>","QuesAnswer":"","QuesParse":"","UseSum":11,"AvgScore":5.0,"ParseAnswerKey":"cdbc5bf0a85d8307cfa1a58c58b18746","QuesAttribute":2},{"ID":7264426,"Guid":"87a80e97-cb72-44f6-b180-c6519fc985be","Title":"2016年全国高中数学联赛四川赛区预赛试题","From":"","Knowledge":"","QuesType":{"ID":2703,"Name":"解答题","IsSelectType":false,"OrderIndex":4,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132862\">解析几何</category><category id=\"132928\">直线与二次曲线方程及性质</category></item>","ChildNum":1,"Time":"2018-12-05T19:50:42","QuesBody":"<div>【题文】已知拋物线<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image177.png\" style=\"vertical-align:middle;\" />过定点C(l,2),在抛物线上任取不同于点C的一点A,直线AC与直线y=x+3交于点P,过点P作x轴的平行线,与抛物线交于点B.<br />(1)证明:直线AB过定点;<br />(2)求△ABC面积的最小值.</div>","QuesAnswer":"","QuesParse":"","UseSum":20,"AvgScore":5.0,"ParseAnswerKey":"fe34667f32421b75490ec29faa346178","QuesAttribute":2},{"ID":7264427,"Guid":"ace329dd-83b5-471d-93db-8cc5c2584c75","Title":"2016年全国高中数学联赛四川赛区预赛试题","From":"","Knowledge":"","QuesType":{"ID":2703,"Name":"解答题","IsSelectType":false,"OrderIndex":4,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":4,"DifValue":0.4,"Name":"较难"},"Categories":"<item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132871\">导数与极限</category><category id=\"133053\">导数的应用</category></item><item><category id=\"27925\">高中数学综合库</category><category id=\"132454\">竞赛知识点</category><category id=\"132871\">导数与极限</category><category id=\"133053\">导数的应用</category><category id=\"133078\">判断单调性</category></item>","ChildNum":1,"Time":"2018-12-05T19:50:42","QuesBody":"<div>【题文】已知a为实数,函数<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image200.png\" style=\"vertical-align:middle;\" />.请讨论函数<img src=\"Upload/2018-12/05/e16d55bf-02da-4795-a2b9-9f262ce2b30e/paper.files/image013.png\" style=\"vertical-align:middle;\" />单调性.</div>","QuesAnswer":"","QuesParse":"","UseSum":12,"AvgScore":5.0,"ParseAnswerKey":"78fd6b48d0e62fdadc7c84109dde1cc3","QuesAttribute":2}],"bankId":0,"PaperQuesCount":16,"ReadSum":88,"schoolId":0}