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## IMO练习曲(附：不等式选摘：Inequalities proposed in Crux Mathematicorum）

(2014-07-16 15:54:51)

### 传播

IMOT 2011                                       1998伊朗数学奥林匹克决赛试题

 problem 5  night 1

http://www.artofproblemsolving.com/community/c6h599319p3557298

Netherlands IMO Team Selection Tests 2014

Dutch IMO TST I Problem 3

mathwizarddude不等式

# IMO练习曲部分试题解答1

http://wuling.name/archives/2619

### 1998年伊朗数学奥林匹克决赛试题(英文版) -

宋庆IMO练习曲第3题

1：宋庆 IMO练习曲

## 2：熊昌进 2014西班牙奥林匹克第2题的证明

【1528*】Let . Then the equation of the tangent line of at is given by . To verify this, note that the slope of the tangent line at is , where . As , we get the desired equation.

Now, consider the function for . Then . Note that the equation has exactly one positive real root by Descartes' Rule of Signs. Thus, for .

It follows that if and , then

Now, suppose that at least one of is less than . WLOG, let be fixed. Set , so .
Since , the point of inflection of is , where , which is approximately . Also, the tangent line at is given by . Since , the tangent line at is above the curve. Furthermore, one can verify that the tangent line at passes through the origin. Since , the tangent line at is still above when . Therefore, we have . It follows that

It suffices to show that , where . We have

and because has a root between and , with and , we have for .
Therefore, the inequality is true and equality holds iff .

（改编自宋庆老师一个猜想）

1. 已知  是正数. 求证:

problem 1 night 4

2. 已知  是实数. 求证：  等价于

problem 7 night 3

3.已知   且  求证:

problem 4 night 10

4. 已知 是满足  的正数 求证｛

problem 7 night 8

5.已知 是满足的正数 ，求证 :

problem 1 night 5

6.已知 两两不同, 均不为零，且 求证：

problem 2 night 5

 7.已知  为三角形三边长. 求证 : a.   为一三角形三边 长 b.             problem 1 night 3

8.已知  为三角形三边长，且  . 求  的最小值

problem 7 night

9.已知  是非负整数，且    ，  .求证

problem 2 night 1

10.已知   的任意一个排列.求证:

problem 6 night 5

11.求证 的小数部分不大于                                                                  problem 6 night 3
12.求证： for                                                         problem 3 night 11
13.已知  ∈  求证：

problem  night 1
已知  ，   是有理数，且  证明 是有理数.
Dutch IMO TST I Problem 3

1.   已知    是满足 的有理数, 证明 是一个有理数.                                                                                                                                            OME 2014 2
2.  已知  是一个形如的整数集合, 其中   ，  是不同的整数.

i) 证明  中任何两个元素的积仍是   的元素.

ii) Determine, reasonably, if there exist infinite pairs of integers  so that but

已知 ，求证：.

mathwizarddude不等式

已知  ，且 .求证
.

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