名校
1 . 设
,函数
.
(1)已知
,求证:函数
为定义域上的奇函数;
(2)已知
.
(i)判断并证明函数
的单调性;
(ii)函数
在区间
上的值域是
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1e69392d21261afd8e5e5f096634669.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5029bd373d0a619fd342eeb67a03dd2e.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e10e1c43b86a8cd4360ca9b57232164.png)
(i)判断并证明函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(ii)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/711e45f600c091e6830c0b70cd012ca3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16ff0183cc03b2dd1262139df3b646b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8573eecbc29f522671b3892ec406c50b.png)
您最近一年使用:0次
名校
2 . 函数
对任意的实数
,有
,当
时,有
.
(1)判断奇偶性并证明.
(2)求证:
在
上为增函数.
(3)若
,解不等式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b6894e8c345a035e89ec672503a01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/280860dd039e1305a5ccc455f63e8223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344655de7e7d632fb819ba1344ab9872.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6000b174147cec2de26041837aec1b3.png)
(1)判断奇偶性并证明.
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b6894e8c345a035e89ec672503a01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e84e89ade5594c4665e43a320ca9f21.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/737818f97b6740bb592d0231b89a1810.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ad6ef4e5005697beb0694cc5759d8e8.png)
您最近一年使用:0次
3 . 若函数
对任意的
,均有
,则称函数
具有性质
.
(1)判断下面两个函数是否具有性质
,并说明理由.①
;②
.
(2)若函数
具有性质
,且
,求证:对任意
有
;
(3)在(2)的条件下,是否对任意
均有
.若成立给出证明,若不成立给出反例.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c75190e49deb89c5a43eda6083422418.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)判断下面两个函数是否具有性质
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abbe323771bc92bf5767e1bd9a46b946.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c904567c3b3734e1eca8d042ef7a7b2d.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25967a292e2486b6406e60447cfb0e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b880d2aa081b381aabbc1634dcf26c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6db419cf1fca9e54646e150752cd7a82.png)
(3)在(2)的条件下,是否对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68f5762af19bbe5d56474384277a5d98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6db419cf1fca9e54646e150752cd7a82.png)
您最近一年使用:0次
2020-05-08更新
|
975次组卷
|
6卷引用:2011届北京市西城区高三二模试卷数学(文科)
(已下线)2011届北京市西城区高三二模试卷数学(文科)北京市首师大附2017-2018学年高三十月月考数学(文)试题2020届上海市上海中学高三下学期高考模拟(4月)数学试题(已下线)重难点12 选考系列(参数方程与不等式)-2021年高考数学【热点·重点·难点】专练(上海专用)(已下线)第四章 指数函数与对数函数单元检测卷(能力挑战)-【一堂好课】2021-2022学年高一数学上学期同步精品课堂(人教A版2019必修第一册)北京市第八十中学2022-2023学年高一上学期期中考试数学试题
解题方法
4 . 已知函数
(其中
),
(1)试判断并证明函数
的单调性;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92cf6f1cf39478c7b037aa45b5d89468.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd8dc2c7954a61c17ce444232f965ab5.png)
(1)试判断并证明函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a34e7e4fb503c94e0008d1fc7a561f6.png)
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名校
5 . 已知函数
其反函数为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
(1)求证:对任意
都有
,对任意
都有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c70f029102bd0b5e762717c3889671fb.png)
(2)令
,讨论
的定义域并判断其单调性(无需证明).
(3)当
时,求函数
的值域;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bb2fb6043949ffd4a0fc14967e23c90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
(1)求证:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dcbca3478eae63853d2aab5332e2e56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0b8e9b3f07d91da4d256d18df240fe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edae93ec9de65d7e8afd2a53063c8ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c70f029102bd0b5e762717c3889671fb.png)
(2)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57a76b586e289841016c49819b99559f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd0f5e152398772be9ec9555664a6407.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ee2ec7a69d7d1f401e04afd231f6515.png)
您最近一年使用:0次
6 . 请仔细阅读以下材料:
已知
是定义在
上的单调递增函数.
求证:命题“设
,若
,则
”是真命题.
证明:因为
,由
得
.
又因为
是定义在
上的单调递增函数,
于是有
. ①
同理有
. ②
由①+ ②得
.
故,命题“设
,若
,则
”是真命题.
请针对以上阅读材料中的
,解答以下问题:
(1)试用命题的等价性证明:“设
,若
,则:
”是真命题;
(2)解关于
的不等式
(其中
).
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a141e697b1a31a9a4e759984e899a5.png)
求证:命题“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/b2a8db4930d64746bee7acb58118f1be.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/add4ea370e3946109d077624424d5f04.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/473303324fc54d9fbef44f60c383cdd4.png)
证明:因为
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/b2a8db4930d64746bee7acb58118f1be.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/add4ea370e3946109d077624424d5f04.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/8e1dba6af48b4f02a02353cfceac54bc.png)
又因为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a141e697b1a31a9a4e759984e899a5.png)
于是有
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/e138ae2d9d174247aa79ca4be523361f.png)
同理有
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/6e01071f3f38469e8e15c3d76700b775.png)
由①+ ②得
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/473303324fc54d9fbef44f60c383cdd4.png)
故,命题“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/b2a8db4930d64746bee7acb58118f1be.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/add4ea370e3946109d077624424d5f04.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/473303324fc54d9fbef44f60c383cdd4.png)
请针对以上阅读材料中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)试用命题的等价性证明:“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/b2a8db4930d64746bee7acb58118f1be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c36ce14a18f423fcff11def7512150e.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/add4ea370e3946109d077624424d5f04.png)
(2)解关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d4c9d254df7fc5169fe8e745a3b74b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5c0c6e3ada0970f9a1fefd7200ff677.png)
您最近一年使用:0次
名校
7 . 已知奇函数
.
(1)试确定
的值;
(2)判断
的单调性,并证明;
(3)若方程
在
上有解,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18ee4a52efc303a9e20f8dc2a2fcbda4.png)
(1)试确定
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9c0d827ef8598ba6b70b34b2bdcd1e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad2edd8edcb21bd41584daf9bb95a5c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5631f3882775ce7c67fc2408b750c503.png)
您最近一年使用:0次
名校
8 . (1)请根据对数函数
来指出函数
的基本性质(结论不要求证明),并画出图象;
(2)拉普拉斯称赞对数是一项“使天文学家寿命倍增”的发明.对数可以将大数之间的乘除运算简化为加减运算,请证明:
;
(3)2017年5月23日至27日,围棋世界冠军柯洁与DeepMind公司开发的程序“AlphaGo”进行三局人机对弈,以复杂的围棋来测试人工智能.围棋复杂度的上限约为
,而根据有关资料,可观测宇宙中普通物质的原子总数约为
.甲、乙两个同学都估算了
的近似值,甲认为是
,乙认为是
.现有两种定义:
![](https://img.xkw.com/dksih/QBM/2017/10/10/1792365589995520/1793820237086720/STEM/2c179cd798bc431f94b813641b8a2aec.png?resizew=190)
①若实数
满足
,则称
比
接近
;
②若实数
,且
,满足
,则称
比
接近
;请你任选取其中一种定义来判断哪个同学的近似值更接近
,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3808314697be51e2ff72179fb6556374.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1b8eea00910cad6239a29a85991f925.png)
(2)拉普拉斯称赞对数是一项“使天文学家寿命倍增”的发明.对数可以将大数之间的乘除运算简化为加减运算,请证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f8734005d48f1e77eeaebe832058dee.png)
(3)2017年5月23日至27日,围棋世界冠军柯洁与DeepMind公司开发的程序“AlphaGo”进行三局人机对弈,以复杂的围棋来测试人工智能.围棋复杂度的上限约为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4f610f3ed73c02ddc0fd21b34d12ee2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/013492f5b66a5b5b9169222c524474b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57584805a70c17d752bbd0def995accc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a0b4b553124abf972a92af238b80480.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d10e3a8a2bd5a1fd799f8640d31d826d.png)
![](https://img.xkw.com/dksih/QBM/2017/10/10/1792365589995520/1793820237086720/STEM/2c179cd798bc431f94b813641b8a2aec.png?resizew=190)
①若实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2e647c14561826ba9e396acc5a3792c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b782dd2de9c9caa840838cd63d817de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
②若实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/413915a68960106812e6577dedac2f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/376218b0e2c4b0bc42f54573c5703a8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5892a5def700f49245c7389aae50a68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57584805a70c17d752bbd0def995accc.png)
您最近一年使用:0次
解题方法
9 . 设函数
定义在
上,对于任意实数
,
,恒有
,且当
时,
.
(1)求
的值.
(2)求证:对任意的
,有
.
(3)证明:
在
上是减函数.
(4)设集合
,
,且
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ac82501b461d044f78e7ae5b86cd3c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5456d544e2f8d22c08f3ccee002dad4a.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f54b6a060d6c51a328341df76013bd89.png)
(2)求证:对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/018857ec6e498113b3b12a730d9313da.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
(4)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34f4d43634c9b34cf3a7aa58ef9f68a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0061c00ac16b749237aebb6c55a4257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dea9a4259cca10c1f5af28e621ebafd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
10 . 请仔细阅读以下材料:
已知
是定义在
上的单调递增函数.
求证:命题“设
,若
,则
”是真命题.
证明 :因为
,由
得
.
又因为
是定义在
上的单调递增函数,
于是有
. ①
同理有
. ②
由①+ ②得
.
故,命题“设
,若
,则
”是真命题.
请针对以上阅读材料中的
,解答以下问题:
(1)试用命题的等价性证明:“设
,若
,则:
”是真命题;
(2)解关于
的不等式
(其中
).
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a141e697b1a31a9a4e759984e899a5.png)
求证:命题“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/24e4835206fe4a69b03e5c5562294155.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/8209882c950f4c02a7aa91c6ad4584ae.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/25072c5bb7274310b540c233b24508ed.png)
证明 :因为
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/24e4835206fe4a69b03e5c5562294155.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/8209882c950f4c02a7aa91c6ad4584ae.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/63e0cee9c0994ebe9e8c9162a5fd4c58.png)
又因为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a141e697b1a31a9a4e759984e899a5.png)
于是有
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/816123fe83654315b9ad464cbbd7d4fd.png)
同理有
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/4cb2879dd5224072a6253ea98f00a84c.png)
由①+ ②得
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/25072c5bb7274310b540c233b24508ed.png)
故,命题“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/24e4835206fe4a69b03e5c5562294155.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/8209882c950f4c02a7aa91c6ad4584ae.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/25072c5bb7274310b540c233b24508ed.png)
请针对以上阅读材料中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)试用命题的等价性证明:“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/24e4835206fe4a69b03e5c5562294155.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c36ce14a18f423fcff11def7512150e.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/8209882c950f4c02a7aa91c6ad4584ae.png)
(2)解关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d4c9d254df7fc5169fe8e745a3b74b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eda48853e8bdb7e266370b4e0d5a258.png)
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