2006年全国高中数学联赛试题及解答

熟悉建筑节能设计标准中有关暖通空调制冷部分、暖通空调制冷设备产品标准中设计选用部分、环境保护及卫生标准中有关本专业的规定条文。掌握上述标准中有关本专业的强制性条文。constructionworkareaintheformofconferencereviews.(2)reviewprogramreviewdateisdeterminedbytheprojectmanager;AccordingtotheprojectmanagerofprojectqualitycontrolDepartmentrequirespreparationofamanagementplan,representativeofthemanagementauditoftheproject,approvedbytheprojectmanager;ReviewProjectOfficeissuednoticesofmeetings;Projectmanagerfacilitatesthereviewmeeting,scheduledcontentunderreview;QualitycontrolDepartmentisresponsiblefortherecords,thepreparationofthereviewreportandcorrectiveactionplan,representativeofthemanagementauditoftheproject,approvedbytheprojectmanager;QualitycontrolDepartmentisresponsibleforthepreparationofrecordsandreportsthedistributionlist,orpublishtotheInternetbyengineeringpartoftheprojectstaffanddepartments.OrganizedbythequalitycontrolDepartmentisresponsiblefortrackingtheimplementationofcorrectivemeasuresinspections.3.4.5theoutputfromthemanagementreviewshallincludethefollowingelements:theeffectivenessandsuitabilityofthequalitymanagementsystem,whetheritbemaintainedandrevised;Qualitypolicyandqualityobjectivesareachievedandtomeetcustomerrequirements,whetheritbemaintainedandrevised;Theallocationofresourcestomeetrequirements,whetheryouneedtoaddoradjust.3.4.6correctiveactiontrackingandverificationontheissuesraisedinthereview,bytheprimaryresponsibilityofdepartmentsdrawupcorrespondingspecificcorrective,preventiveandcorrectivemeasures,andseriouslyimplemented.ProjectqualitycontrolDepartmentisresponsibleforthetracking,verificationandevaluationtheeffectofcorrectiveandpreventiveactions.3.4....(1)thesafetyprogramsanddepartmentsdealingwiththeformationofurbanspatial2006年全国高中数学联合竞赛试题参考答案及评分标准一、选择题(本题满分36分，每小题6分)1．答C．解：令∠ABC＝α，过A作AD⊥BC于D，由||→BA－t→BC≥||→AC，推出||→BA2－2t→BA·→BC＋t2||→BC2≥||→AC2,令t＝→BA·→BC||→BC2，代入上式，得||→BA2－2||→BA2cos2α＋||→BA2cos2α≥||→AC2，即||→BA2sin2α≥||→AC2,也即||→BAsinα≥||→AC．从而有||→AD≥||→AC．由此可得∠ACB＝π2．2．答B．解：因为x＞0，x≠12x2＋x－1＞0，解得x＞12且x≠1．由logx(2x2＋x－1)＞logx2－1，logx(2x3＋x2－x)＞logx20＜x＜1，2x3＋x2－x＜2或x＞1，2x3＋x2－x＞2．解得0＜x＜1或x＞1．所以x的取值范围为x＞12且x≠1．3答C．解：5x－a≤0x≤a5；6x－b＞0x＞b6．要使A∩B∩N＝{2，3，4}，则1≤b6＜2，4≤a5＜5，即6≤b＜12，20≤a＜25．所以数对(a，b)共有C61C51＝30个．4．答A．解：建立直角坐标系，以A为坐标原点，AB为x轴，AC为y轴，AA1为z轴，则F(t1，0，0)(0＜t1＜1)，E(0，1，12)，G(12，0，1)，D(0，t2，0)(0＜t2＜1)．所以→EF＝(t1，－1，－12)，→GD＝(－12，t2，－1)．因为GD⊥EF，所以t1＋2t2＝1，由此推出0＜t2＜12．又→DF＝(t1，－t2，0)，||→DF＝t12＋t22＝5t22－4t2＋1＝5(t2－25)2＋15，从而有15≤||→DF＜1．5．答A．解：显然f(x)＝x3＋log2(x＋x2＋1)为奇函数，且单调递增．于是若a＋b≥0，则a≥－b，有f(a)≥f(－b)，即f(a)≥－f(b)，从而有f(a)＋f(b)≥0．反之，若f(a)＋f(b)≥0，则f(a)≥－f(b)＝f(－b),推出a≥－b，即a＋b≥0．6．答B．熟悉建筑节能设计标准中有关暖通空调制冷部分、暖通空调制冷设备产品标准中设计选用部分、环境保护及卫生标准中有关本专业的规定条文。掌握上述标准中有关本专业的强制性条文。constructionworkareaintheformofconferencereviews.(2)reviewprogramreviewdateisdeterminedbytheprojectmanager;AccordingtotheprojectmanagerofprojectqualitycontrolDepartmentrequirespreparationofamanagementplan,representativeofthemanagementauditoftheproject,approvedbytheprojectmanager;ReviewProjectOfficeissuednoticesofmeetings;Projectmanagerfacilitatesthereviewmeeting,scheduledcontentunderreview;QualitycontrolDepartmentisresponsiblefortherecords,thepreparationofthereviewreportandcorrectiveactionplan,representativeofthemanagementauditoftheproject,approvedbytheprojectmanager;QualitycontrolDepartmentisresponsibleforthepreparationofrecordsandreportsthedistributionlist,orpublishtotheInternetbyengineeringpartoftheprojectstaffanddepartments.OrganizedbythequalitycontrolDepartmentisresponsiblefortrackingtheimplementationofcorrectivemeasuresinspections.3.4.5theoutputfromthemanagementreviewshallincludethefollowingelements:theeffectivenessandsuitabilityofthequalitymanagementsystem,whetheritbemaintainedandrevised;Qualitypolicyandqualityobjectivesareachievedandtomeetcustomerrequirements,whetheritbemaintainedandrevised;Theallocationofresourcestomeetrequirements,whetheryouneedtoaddoradjust.3.4.6correctiveactiontrackingandverificationontheissuesraisedinthereview,bytheprimaryresponsibilityofdepartmentsdrawupcorrespondingspecificcorrective,preventiveandcorrectivemeasures,andseriouslyimplemented.ProjectqualitycontrolDepartmentisresponsibleforthetracking,verificationandevaluationtheeffectofcorrectiveandpreventiveactions.3.4....(1)thesafetyprogramsanddepartmentsdealingwiththeformationofurbanspatial解：出现奇数个9的十进制数个数有A＝C2006192005＋C2006392003＋…＋C200620059．又由于(9＋1)2006＝k＝0Σ2006C2006k92006－k以及(9－1)2006＝k＝0Σ2006C2006k(－1)k92006－k从而得A＝C2006192005＋C2006392003＋…＋C200620059＝12(－82006)．二、填空题(本题满分54分，每小题9分)7.填[0，98]．解：f(x)＝sin4x－sinxcosx＋cos4x＝1－12sin2x－12sin22x．令t＝sin2x，则f(x)＝g(t)＝1－12t－12t2＝98－12(t＋12)2．因此－1≤t≤1ming(t)＝g(1)＝0，－1≤t≤1maxg(t)＝g(－12)＝98．故，f(x)∈[0，98]．8.填[－55，55]．解：依题意，得|z|≤2(a＋cosθ)2＋(2a－sinθ)2≤42a(cosθ－2sinθ)≤3－5a2．－25asin(θ－φ)≤3－5a2(φ＝arcsin55)对任意实数θ成立．25|a|≤3－5a2|a|≤55，故a的取值范围为[－55，55]．9．填3－1．.解：由平面几何知，要使∠F1PF2最大，则过F1，F2，P三点的圆必定和直线l相切于点P．直线l交x轴于A(－8－23，0)，则∠APF1＝∠AF2P，即∆APF1∽∆AF2P，即|PF1||PF2|＝|AP||AF2|⑴又由圆幂定理，|AP|2＝|AF1|·|AF2|⑵而F1(－23，0)，F2(23，0)，A(－8－23，0)，从而有|AF1|＝8，|AF2|＝8＋43．代入⑴，⑵得，|PF1||PF2|＝|AF1||AF2|＝88＋43＝4－23＝3－1．10．填