### Example Problem: Inflection Point Offset

Source: https://github.com/raineblog/whk/blob/main/docs/science/function/4.md

This section presents an example of using the inflection point offset method to prove an inequality.

```mathematics
例题：已知函数 $f(x)=2\ln x+x^2-1$，若 $x_1,x_2$ 是两个不相等的正数，且 $f(x_1)+f(x_2)=0$，证明 $x_1+x_2>2$。

$$f'(x)=\dfrac{2}{x}+2x\ge4$$

因此 $f(x)$ 单调递增，且 $f(1)=0$，注意到 $2$ 就是两倍的零点，我们不妨设

$$0<x_1<1<x_2<2$$

因为 $x_2>2$ 的时候，显然 $x_1+x_2>2$。考虑这个部分，即证

$$f(x_2)>f(2-x_1)$$

又因为 $f(x_2)=-f(x_1)$，因此证

$$-f(x_1)>f(2-x_1)$$

不妨设

$$g(x)=f(x)+f(2-x)=2\ln(2x-x^2)-2(2x-x^2)+2$$

其中 $x\in(0,1)$，不妨令 $t=2x-x^2\in(0,1)$，由 $\ln t<x-1$，不等式显然成立。
```

--------------------------------

### Example Problem: Finding the range of MN length

Source: https://github.com/raineblog/whk/blob/main/docs/science/geometry/2.md

This is an example problem demonstrating how to find the range of the length of line segment MN, which involves geometric concepts and coordinate calculations.

```markdown
```md {admonition=note title="例题"}
已知 $M,N$ 为圆 $x^2+y^2=9$ 上两点，点 $A(1,2)$，且 $AM
AN$，则线段 $MN$ 的长的取值范围是？

我们发现只需要求出 $|OP|$ 的取值范围即可：

$$ 
\paren{\dfrac{|MN|}{2}}^2+|OP|^2=|OM|^2
$$

我们不妨取坐标原点 $O$，$MN$ 中点 $P$，容易发现方程：

$$ |OP|^2+|AP|^2=9 $$

不妨设 $P(x,y)$，解得：

$$ 
\paren{x-\dfrac12}^2+\paren{y-1}^2=\dfrac{13}4
$$

即可得到 $|OP|$ 的取值范围。
```
```

--------------------------------

### Simple Examples

Source: https://github.com/raineblog/whk/blob/main/dev/finished/半缩醛.md

Examples of acetal and ketal formation.

```chemical
乙醛 (CH₃CHO) + 2 CH₃OH → CH₃CH(OCH₃)₂（乙醛二甲基缩醛）
```

```chemical
丙酮 + 乙二醇 → 2,2-二甲基-1,3-二氧戊环（常见缩酮保护基）
```

--------------------------------

### Claisen-Schmidt Condensation Example

Source: https://github.com/raineblog/whk/blob/main/dev/finished/有机拓展.md

Example of Claisen-Schmidt condensation between benzaldehyde and acetone.

```latex
$C_6H_5CHO + CH_3COCH_3 \rightarrow C_6H_5CH=CHCOCH_3$
```

--------------------------------

### Example Problem: Constructing a function to prove an inequality

Source: https://github.com/raineblog/whk/blob/main/docs/science/function/4.md

This section demonstrates how to use the construction function method to solve a problem involving finding extreme points and proving an inequality.

```mathematics
例题：已知函数 $f(x)=\dfrac{1}{2}x+m+\dfrac{3}{2x}-\ln x(m\in\R)$，若 $x_1,x_2$ 是函数 $g(x)=xf(x)$ 的两个极值点，且 $x_1<x_2$，证明 $x_1x_2<1$。

我们令 $g(x)=xf(x)=mx-x\ln x+\dfrac{1}{2}x^2+\dfrac{3}{2}$，对其求导

$$g'(x)=x-\ln x+m-1$$

$$g''(x)=\dfrac{x-1}{x}$$

当 $0<x<1$ 时，$g''(x)<0$，$g'(x)$ 单调递减；当 $x>1$ 时，$g''(x)>0$，$g'(x)$ 单调递增。因此 $1$ 为 $g'(x)$ 的极值点，因此 $0<x_1<1<x_2$。

要证 $x_1x_2<1$ 即证

$$x_2<\dfrac{1}{x_1}$$

因为 $0<x_1<1$ 所以 $\dfrac{1}{x_1}>1$，因此尝试证明

$$g'(x_2)<g'\left(\dfrac{1}{x_1}\right)$$

又因为 $g'(x_1)=g'(x_2)=0$，因此证明

$$g'(x_1)<g'\left(\dfrac{1}{x_1}\right)$$

令函数

$$h(x)=g'(x)-g'\left(\dfrac{1}{x}\right)=x-\dfrac{1}{x}-2\ln x(0<x<1)$$

求导得

$$h'(x)=\dfrac{(x-1)^2}{x^2}>0$$

因此 $h(x)$ 单调递增，从而 $h(x)<h(1)=0$，得证。

这道题也可以用构造变量法，同样令 $t=\dfrac{x_2}{x_1}>1$，证明

$$\ln t<\sqrt{t}-\dfrac{1}{\sqrt{t}}$$

即可。
```

--------------------------------

### Example 1: Divisibility by 6

Source: https://github.com/raineblog/whk/blob/main/docs/science/number/3.md

Demonstrates the use of modular arithmetic to prove that d is divisible by 6, where d = a^1999 + b^1999 + c^1999 and a + b + c = 0.

```mathematics
u^1999 === u(mod 2) implies d === a + b + c === 0(mod 2) implies 2|d
```

```mathematics
u^3 === u(mod 3) implies u^1999 === u(mod 3)
```

--------------------------------

### Example of One-Carbon Growth via Nitrile Hydrolysis

Source: https://github.com/raineblog/whk/blob/main/dev/finished/有机拓展.md

Example showing one-carbon growth from acetaldehyde to lactic acid.

```latex
$CH_3CHO \xrightarrow{HCN} CH_3CH(OH)CN \xrightarrow{水解} CH_3CH(OH)COOH$
```

--------------------------------

### 例题

Source: https://github.com/raineblog/whk/blob/main/docs/science/function/3.md

已知 $0<x<1$，则 $\dfrac9x+\dfrac{16}{1-x}$ 的最小值为？ $\dfrac9x+\dfrac{16}{1-x}=\paren{\dfrac9x+\dfrac{16}{1-x}}[(x)+(1-x)]=25+9\dfrac{1-x}x+16\dfrac x{1-x}\ge49$

```markdown
```md {admonition="note" title="例题"}
已知 $0<x<1$，则 $\dfrac9x+\dfrac{16}{1-x}$ 的最小值为？

$$ 
\begin{aligned}
\dfrac9x+\dfrac{16}{1-x}&=\\paren{\\dfrac9x+\\dfrac{16}{1-x}}[(x)+(1-x)] \\
&=25+9\\dfrac{1-x}x+16\\dfrac x{1-x}\ge49
\end{aligned}
$$ 
```
```

--------------------------------

### Sports Narration Examples (Athletics)

Source: https://github.com/raineblog/whk/blob/main/docs/cultural/writing/3.md

Examples of describing actions in track and field events, focusing on speed and explosiveness.

```English
田径运动（侧重速度与爆发）

- 起跑与加速：Sb. bolted/dashed forward, took a deep breath and sprinted like an arrow. (某人箭步冲出，深吸一口气，像箭一样冲刺)。

- 途中状态：With sweat streaming down his face, he struggled to his feet and kept on running. (汗水流过脸庞，他挣扎着站起继续奔跑)。

- 冲刺：He gathered all his strength, lunged forward and crossed the finish line. (他凝聚全身力量，向前猛冲，跨过了终点线)。
```

--------------------------------

### Admonition Syntax Example

Source: https://github.com/raineblog/whk/blob/main/AGENTS.md

Example of using the admonition syntax for notes, tips, warnings, or details.

```markdown
{admonition="note" title="注意"}
内容支持 Markdown。

```

--------------------------------

### 例题

Source: https://github.com/raineblog/whk/blob/main/docs/science/function/3.md

已知 $a,b>0$ 且 $ab=a+b+3$，则 $ab,a+b$ 的最小值分别为？ 方法一：由 $ab=a+b+3$，得到 $a=\dfrac{b+3}{b-1}$，带入消元即可。 方法二：$ab=a+b+3\ge2\sqrt{ab}+3$，解得 $\sqrt{ab}\ge3$ 即 $ab\ge9$。 方法三：由 $ab-a-b+1=4$ 得 $4=(a-1)(b-1)\le\dfrac14(a+b-2)^2$，则 $a+b\ge6$。

```markdown
```md {admonition="note" title="例题"}
已知 $a,b>0$ 且 $ab=a+b+3$，则 $ab,a+b$ 的最小值分别为？

方法一：由 $ab=a+b+3$，得到 $a=\dfrac{b+3}{b-1}$，带入消元即可。

方法二：$ab=a+b+3\ge2\sqrt{ab}+3$，解得 $\sqrt{ab}\ge3$ 即 $ab\ge9$。

方法三：由 $ab-a-b+1=4$ 得 $4=(a-1)(b-1)\le\dfrac14(a+b-2)^2$，则 $a+b\ge6$。
```
```

--------------------------------

### Add MkDocs Material and Script Submodules

Source: https://github.com/raineblog/whk/blob/main/docs/home/roadmap.md

Commands to add MkDocs Material and MkDocs Script as git submodules, and update them.

```shell
git submodule add https://github.com/raineblog/mkdocs-material material
git submodule add https://github.com/raineblog/mkdocs-script script
git submodule add https://github.com/raineblog/intro docs/intro/
git submodule update --remote
```

--------------------------------

### Sports Narration Examples (Ball Sports)

Source: https://github.com/raineblog/whk/blob/main/docs/cultural/writing/3.md

Examples of describing actions in ball sports, focusing on coordination and turning points.

```English
球类运动（侧重协调与转折）

- 控球与进攻：He gripped the ball tightly, dodged the opponent and made a sudden turn. (他紧紧握球，闪躲对手，突然转身)。

- 关键投射：Sb. took a moment to focus, leaped into the air and shot the ball with precision. (某人专注片刻，跃入空中，精准投球)。
```

--------------------------------

### 例题

Source: https://github.com/raineblog/whk/blob/main/docs/science/function/3.md

已知 $a,b>0$ 且 $2a+b=1$，则 $\dfrac1a+\dfrac ab$ 的最小值为？ $\dfrac1a+\dfrac ab=\dfrac{2a+b}a+\dfrac{a}{b}=2+\dfrac ab+\dfrac ba\ge4$

```markdown
```md {admonition="note" title="例题"}
已知 $a,b>0$ 且 $2a+b=1$，则 $\dfrac1a+\dfrac ab$ 的最小值为？

$$ 
\dfrac1a+\dfrac ab=\dfrac{2a+b}a+\dfrac{a}{b}=2+\dfrac ab+\dfrac ba\ge4 
$$ 
```
```