1 . 设![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c30ee388d945875e585c25daec7e2fc4.png)
.
(1)若
,求证:
是完全平方数;
(2)证明:存在无穷多个正整数对
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c30ee388d945875e585c25daec7e2fc4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f82b872914335d22f93f6c866d2b5c61.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b2fb1017d5dfe3a300de3e14a71b6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43f41be870e84c819362787849770519.png)
(2)证明:存在无穷多个正整数对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/698c4d4e50062b4a7dd70fe1b4ab4fd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b2fb1017d5dfe3a300de3e14a71b6f.png)
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2 . 如图,
、
为
的两条切线,切点分别为
、
,过点
的直线交
于点
、
,交弦
于点
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/193f204f87cbc609f0a791ec5d1ff572.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/6/4f7c6d4e-9b3b-41b1-bbf6-9bebfc7767fa.png?resizew=187)
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3 . 已知数列
满足
,
.记
.求:
(1)求数列
的通项公式;
(2)求数列
前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86fc336b4a83bf6d66c4afcc431597f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a9e7a4ac9e605c19dfb39116a92b811.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eed845e217cc13e47041f5b45b4e135.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec4bdc2a6d4fc387dc621f0b5a268c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
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4 . 已知函数
,数列
、
满足
,
,
,
,
.
(1)求
的取值范围,使得对任意的正整数
,都有
;
(2)若
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fbd56fe7b6b14439638211d79f79eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/390636a89883bd64bf8da9bf8654aff9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d765033fa3e470b4b4bae90a28514587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbc5ac1542da56851d19240e723a1530.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca42e7cd03df12843de137eb8843f37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66e713f441cb0f03cf7ca5d7f4cd157f.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2798e1dcab1f7f0fe3b8a94b3cd6a88.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b72ddd7de598464a37b10f03f67b904.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb774169e51010958807547268ca0430.png)
您最近一年使用:0次
2009高三·陕西·竞赛
5 . 如图,设点
、
,
内切圆的圆心在直线
上移动.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/3/e207f4f9-0b4d-432b-b682-a2c112eaa2fa.png?resizew=183)
(1)求点
的轨迹方程;
(2)若过点
作两条射线,分别交(1)中所求轨迹于点
、
两点,且
,求证:直线
必过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2e2338cc3b915b80e040dbbcfc52402.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68e2e4abcfc85ff728fbbec8cfad7765.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/3/e207f4f9-0b4d-432b-b682-a2c112eaa2fa.png?resizew=183)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99296bab1b42898e7ca336a822510258.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aeb05a188c6f73606a0e7f1fb6d42e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
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2009高三·陕西·竞赛
6 . 设
,
.若对任意的
,都有
恒成立,试求
、
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eeae4e4940d654f083f1073ba872504.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23241b8f108df2ebb858c6ab622db460.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d868a70648e7bcb20b453c911aec687e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
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2009高三·陕西·竞赛
7 . 把长为
的线段分成三段,这三条线段能构成三角形的概率为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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8 . 一个含有底面的半球形容器内放置有三个两两外切的小球.若这三个小球的半径均为1,且每个小球都与半球的底面和球面相切,则该半球的半径![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e3ef4881bd7c5860178dbdbc7bba6e3.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e3ef4881bd7c5860178dbdbc7bba6e3.png)
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2009高三·陕西·竞赛
名校
9 . 如图,在
中,
,
.若
为
的外心,则
的值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3ad4c0ba3a6750537789844d0ec419d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a3885481fba1f9843e04d0070d6ace7.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/17/4bbad6f0-64ed-4a8f-953b-b54d30859009.png?resizew=134)
您最近一年使用:0次
2018-12-24更新
|
642次组卷
|
4卷引用:2009年全国高中数学联赛陕西赛区预赛试题
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