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已知抛物线

的焦点为
F,过
F的直线
l与
C交于
A,
B两点,点
P在
C的准线上,那么(
)
A.若PA与C相切,则PB也与C相切 |
B. |
C.若点P在x轴上,则 为定值 |
D.若点P在x轴上,且满足 ,则直线l的斜率绝对值为 |
已知数列

为有穷数列,且

,若数列

满足如下两个性质,则称数列

为
m的
k增数列:①

;②对于

,使得

的正整数对

有
k个.
(1)写出所有4的1增数列;
(2)当

时,若存在
m的6增数列,求
m的最小值;
(3)若存在100的
k增数列,求
k的最大值.
若函数

有唯一零点,则

的取值范围为
______.
已知函数

.
(1)若

,讨论

的单调性.
(2)若

有三个极值点

,

,

.
①求

的取值范围;
②求证:

.
数列

满足

则称数列

为下凸数列.
(1)证明:任意一个正项等比数列均为下凸数列;
(2)设

,其中

,

分别是公比为

,

的两个正项等比数列,且

,证明:

是下凸数列且不是等比数列;
(3)若正项下凸数列的前

项和为

,且

,求证:

.
已知函数

.
(1)当

时,求函数

的单调区间和极值;
(2)当

时,不等式

恒成立,求

的取值范围.
已知数列

的前
n项和为

.若对每一个

,有且仅有一个

,使得

,则称

为“
X数列”.记

,

,称数列

为

的“余项数列”.
(1)若

的前四项依次为0,1,

,1,试判断

是否为“
X数列”,并说明理由;
(2)若

,证明

为“
X数列”,并求它的“余项数列”的通项公式;
(3)已知正项数列

为“
X数列”,且

的“余项数列”为等差数列,证明:

.
已知

是边长为1的正六边形边上相异的三点,则

的取值范围是
______.
已知

为实数,用

表示不超过

的最大整数,例如

,对于函数

,若存在

,使得

,则称函数

是“

函数”.
(1)判断函数

是否是“

函数”;
(2)设函数

是定义在

上的周期函数,其最小正周期是

,若

不是“

函数”,求

的最小值;
(3)若函数

是“

函数”,求

的取值范围.