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已知正项数列的前项积为，且满足.
(1)求证：数列为等比数列；
(2)若，求n的最小值.

已知函数.
（1）当时，函数在区间上的最大值为，试求实数 的取值范围；
（2）当时，若不等式对任意）恒成立，求实数 的取值范围．

已知函数，若对任意，有恒成立，则实数的取值范围是______.

对于数列，若存在正数M，使得对一切正整数n，都有，则称数列为有界数列；若这样的正数M不存在，则称数列为无界数列．下列说法正确的有（       ）

A．等比数列的公比为，若，则是有界数列

B．若数列的通项，则是有界数列

C．若正项数列满足：，则是无界数列

D．若数列满足：，且，则是有界数列

阅读下面短文，根据所给情节进行续写，使之构成一个完整的故事。

Last December, Alvin Bamburg, 66, was deep in the woods in Grand Cane, Louisiana, when something caught his eye. As Alvin approached, he saw that the object was a broken balloon. Attached was a piece of paper decorated with sparkly star stickers. It was a child’s Christmas wish list.

“Dear Santa,” the handwritten note read. “My name is Luna. I am four years old. I live in Liberal, Kansas. This year I have been nice. I would like candy, Spider-Man ball, Frozen doll, puppy, My Little Pony. With love, Luna.”

Alvin’s heart hammered in his chest. Ever since he was a child, he had dreamed of this very scene. He believed this was his childhood wish coming true. And he knew he was going to make Luna’s wish come true too. He just wasn’t sure how.

Liberal is more than 650 miles from Grand Cane. But Alvin’s wife was not held back by the distance. As a passionate Facebook user, she had seen people find all sorts of connections through the site and thought it might help her husband find Luna.

Alvin posted a photo of the balloon and the Christmas list on his Facebook page, asking for help locating the sender. At first, he wasn’t confident it would work. But as he saw the number of people sharing his post creep into the hundreds, his hope grew into expectation.

Meanwhile, Leticia Flores, the mother of four-year old Luna had no idea that such an effort was underway. It had been a hard year for Leticia and her girl. As Covid-19 spread, the family felt lonely, scared and isolated. On a particularly tough day last December, Leticia came up with the idea of having the girl send a letter to Santa by balloon.

One day in early January, Leticia noticed that a friend had been calling her all morning. When Leticia finally called her back, the friend blurted, “Someone found Luna’s balloon.”

注意：1. 所续写短文的词数应为150左右；
2. 至少使用5个短文中标有下划线的关键词语；
3. 续写部分分为两段，每段的开头语已为你写好；
4. 续写完成后，请用下划线标出你所使用的关键词语。
Paragraph 1：

At her friend’s urging, she logged on to Facebook and saw Alvin’s post.

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Paragraph 2：

Still, there was one wish on the list that couldn’t go in the box: A puppy.

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假如你是高中生李华，校英文报以“感恩”为主题，向全校学生征文。请你写一篇短文投稿，内容包括：
1. 感恩对象及原因；
2. 你的回报方式。
注意：词数100左右；可以适当增添细节，以使行文连贯。
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

如图甲所示，倾角的斜面固定在水平地面上，斜面上放置长度、质量的长木板A，长木板的下端恰好与斜面底部齐平；一可视为质点、质量、带电量的物块B放在木板上，与木板上端距离；与木板上端距离的虚线右侧存在足够宽匀强电场，电场方向垂直斜面向上．时刻起，一沿斜面向上的恒力F作用在长木板上，后撤去F，物块B在内运动的图像如图乙所示，且物块B在时速度恰好减为0．已知物块与长木板间的动摩擦因数，长木板与斜面间的动摩擦因数，物块B带电量始终不变，重力加速度g取，求：
（1）恒力F的大小；
（2）匀强电场的电场强度大小；
（3）从时刻起到木板下端再次与斜面底部齐平的过程中，物块B与木板A间因摩擦产生的热量．

如图所示光滑圆弧轨道AB固定在水平面上与水平面平滑连接，圆弧轨道最低点A静止放置物块b、c（可看做质点），b、c的质量分别为m、2m，b、c间有少量火药（质量可忽略），某时刻火药燃烧将b、c迅速分开，分开后b以速度向左冲上圆弧，经一段时间再次回到b、c分开位置后继续向右运动，当c刚好停止运动时b与之发生第一次碰撞。已知b与c的所有碰撞均为弹性碰撞，b与水平面间没有摩擦，c与水平面间的动摩擦因数为μ，最大静摩擦力等于滑动摩擦力，重力加速度大小为g。已知该物块b从圆弧底端冲上圆弧到再次回到圆弧底端所用时间与b冲上圆弧速度大小无关，可用圆弧半径R表示为（R为题中未知量）求：
（1）圆弧轨道半径的大小；
（2）b与c发生第3次碰撞前的速度；
（3）b、c第n次碰撞后到第n+1次碰撞前c运动的位移大小（n=l、2、3……）。

太阳方位角指太阳光线在地平面上的投影与当地经线的夹角。太阳方位角以太阳位于正北方向为零，顺时针方向逐渐变大，取值范围是0°～360°。图为75°N某地一年内日出方位角变化。据此完成下面小题。

1．图中，75°N出现极昼的时段是（     ）

A．甲—丙

B．甲—乙

C．丙—丁

D．丁—乙

2．图中甲、乙时刻（     ）

A．太阳高度低于0°	B．太阳高度等于0°
C．太阳从东南升起	D．太阳从西南落下

某兴趣小组设计的连锁机械游戏装置如图所示。左侧有一固定的四分之一圆弧轨道，其末端B水平，半径为3L；在轨道末端等高处有一质量为m的“”形小盒C（可视为质点），小盒C与大小可忽略、质量为3m的物块D通过光滑定滑轮用轻绳相连，左侧滑轮与小盒C之间的绳长为2L；物块D压在质量为m的木板E左端，木板E上表面光滑，下表面与水平桌面间动摩擦因数（最大静摩擦力等于滑动摩擦力），木板E右端到桌子右边缘固定挡板（厚度不计）的距离为L；质量为m且粗细均匀的细杆F通过桌子右边缘的光滑定滑轮用轻绳与木板E相连，木板E与定滑轮间轻绳水平，细杆F下端到地面的距离也为L；质量为0.25m的圆环（可视为质点）套在细杆F上端，环与杆之间滑动摩擦力和最大静摩擦力相等，大小为0.5mg。开始时所有装置均静止，现将一质量为m的小球（可视为质点）从圆弧轨道顶端A处由静止释放，小球进入小盒C时刚好能被卡住（作用时间很短可不计），此时物块D对木板E的压力刚好为零。木板E与挡板相撞、细杆F与地面相撞均以原速率反弹，最终圆环刚好到达细杆的底部。不计空气阻力，重力加速度为g，求：
（1）小球与小盒C相撞后瞬间，小盒C的速度；
（2）小球在四分之一圆弧轨道上克服摩擦力所做的功；
（3）木板E与挡板碰后，向左返回的最大位移；
（4）细杆F的长度。