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《2018年北京大学保送生数学真题及答案》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
1、2012年北京大学保送生考试数学试题及参考答案1.已知数列为正项等比数列,且,求的最小值.解:设数列的公比为,则,.由知.,当且仅当即时,有最小值.2.已知为二次函数,且成正项等比数列,求证:.证法一:设,数列的公比为,则,①②③①②得,②③得.若,则;若,则与矛盾..证法二:由成等比数列得,.三点满足,三点共线,与三点在抛物线上矛盾,.3.称四个顶点都落在三角形三边上的正方形叫三角形的内接正方形.若锐角三角形的三边满足,证明:这个三角形的内接正方形边长的最小值为.解:如图所示,设正方形的边长为,,,.同理可得其它两用人才种情况下内接正方形边长为.,
2、,这个三角形的内接正方形边长的最小值为.4.从点发出两条射线,已知直线交于两点,且(为定值),记中点为,求证:的轨迹为双曲线.解:以的角平分线所在直线为轴建立如图所示的直角坐标系.设,,,则,.,得,的轨迹为双曲线.5.已知满足,求证:,使.证明:用反证法,假设,.令,则,且.与矛盾,,使.内部资料,请勿外传!9JWKffwvG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmA&UE9aQ@Gn8xp$R#͑Gx^Gjqv^$UE9wEwZ#Qc@UE%&
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