From 04ef90111595e246b8f62ee5193f1dfd36398703 Mon Sep 17 00:00:00 2001 From: lmj01 Date: Thu, 30 May 2024 17:10:14 +0800 Subject: [PATCH] update --- articles/demo.md | 19 +++++- cg/image/index.md | 6 +- exercises/index.md | 4 ++ exercises/math.primary.md | 20 ++++--- exercises/math.secondary.md | 74 +++++++++++++++++++++++- exercises/quadratic.equation.md | 10 +++- images/geogebra/geogebra-2024-5-30.png | Bin 0 -> 13961 bytes images/geogebra/geogebra-2024-5-30a.png | Bin 0 -> 17939 bytes 8 files changed, 119 insertions(+), 14 deletions(-) create mode 100644 images/geogebra/geogebra-2024-5-30.png create mode 100644 images/geogebra/geogebra-2024-5-30a.png diff --git a/articles/demo.md b/articles/demo.md index 171256b..3e394ef 100644 --- a/articles/demo.md +++ b/articles/demo.md @@ -1,6 +1,10 @@ -# markdown demo -> markdown渲染成HTML的插件 +# [markdown](https://commonmark.org/) +[中文文档](https://markdown.com.cn/) + +markdown渲染成HTML的插件 + +如果不方便表示的,可以直接使用HTML5的标签来表示内容 ```js function func(a, b) { @@ -10,6 +14,10 @@ function func(a, b) { [What's In A Class?] [1] +
+alert + + ## [alert](https://github.com/bent10/marked-extensions/tree/main/packages/alert) - [Enables GFM alerts](https://github.com/orgs/community/discussions/16925) @@ -34,6 +42,8 @@ function func(a, b) { > **Warning** > This is a warning +
+ ## table - **table one** @@ -64,6 +74,10 @@ function func(a, b) { | Header | Title | Here's this | | Paragraph | Text | And more | + +
+katex + ## [katex](https://katex.org/) - [js katex api ](https://katex.org/docs/api) @@ -225,6 +239,7 @@ $$ [maths symbols for latex](https://mirrors.jlu.edu.cn/CTAN/info/symbols/math/maths-symbols.pdf) +
## 其他 diff --git a/cg/image/index.md b/cg/image/index.md index c4975ca..a6025cc 100644 --- a/cg/image/index.md +++ b/cg/image/index.md @@ -9,4 +9,8 @@ - [tiff](/cg/image/tiff.md) -- [ImageMagick](/cg/image/imageMagick.md) \ No newline at end of file +- [ImageMagick](/cg/image/imageMagick.md) + +## 压缩 + +- [Caesium is an image compression software that helps you store, send and share digital pictures, supporting JPG, PNG and WebP formats. You can quickly reduce the file size (and resolution, if you want) by preserving the overall quality of the image. ](https://github.com/Lymphatus/caesium-image-compressor) \ No newline at end of file diff --git a/exercises/index.md b/exercises/index.md index c02299a..8aeba8b 100644 --- a/exercises/index.md +++ b/exercises/index.md @@ -17,3 +17,7 @@ ## 物理学 - [物理学](/exercises/physics.md) + +## 工具 + +- [Geogebra几何平面作图在线工具,用meijie.lmj@outlook.com登录的](https://www.geogebra.org/geometry?lang=zh_CN) \ No newline at end of file diff --git a/exercises/math.primary.md b/exercises/math.primary.md index f615916..eaf362b 100644 --- a/exercises/math.primary.md +++ b/exercises/math.primary.md @@ -2,7 +2,8 @@ ## 模数 -### 2024-5-23 +
+2024-5-23 $$ A = 1 \times 3 \times 5 \times 7 \times 9 \times 11 \times \cdots \cdots \times 2023 \text{的末两位是多少?} @@ -47,7 +48,11 @@ $$ 所以答案是末三位数是625 -### 2024-5-24剩余定理 +
+ +
+2024-5-24剩余定理 + 就是组合数学中的同余概念 @@ -71,9 +76,12 @@ $ x \equiv (2 \times 70 + 4 \times 21 + 6 \times 15) \pmod 105 = 104 \to 104 + 9 可参考[数论笔记](/exercises/number.theory.md)中的关于同余定理。 +
+ ## 组合 -### 2024-5-24 +
+2024-5-24 一个自然数经过不同的排列,可以组成不同的多位数,例如102,共有4中不同的排列方法。 102,120,201,210。那问有9种排列方法的4位数共有多少个呢? @@ -97,8 +105,4 @@ $ x \equiv (2 \times 70 + 4 \times 21 + 6 \times 15) \pmod 105 = 104 \to 104 + 9 这种题型也是慢慢地分析,分类后一一判断。本题也可以拓展,9种排列改成12,6等。 - - - - - +
diff --git a/exercises/math.secondary.md b/exercises/math.secondary.md index fa4a677..efae6a3 100644 --- a/exercises/math.secondary.md +++ b/exercises/math.secondary.md @@ -2,9 +2,79 @@ - 平面几何 +## 几何 + +
+2024-5-30 + +$\text{已知} AB=AC=5,BC=6,BD=AE, AF \perp DE,\text{求解}\frac{AF}{DE}=?$ + +![2024-5-30](/images/geogebra/geogebra-2024-5-30.png '图1') + +因为题目只是求值,提示采用特殊值法,设点D是AB的中点,依题意E也是中点,即DE是等腰三角形ABC的中位线,DE=0.5BC=3,此时AF是三角形ABC的中垂线 +BF=3,根据毕达哥拉斯定理,AF=4, 即结果为 + +$\frac{AF}{DE}=\frac{4}{3}$ + +如果是证明题的化,如何处理呢?特殊值法时类似的思路,当D是不是中点时,始终有AF垂直于DE,随着D点移动时F也跟随移动,旋转出来两个角。 + +要显示两个角,就需要三角形,作辅助线如下:作三角形ABC的垂线AM,过D点作BC的平行线DP,过E点作BC的垂线,交DP延长线于G。EDG和FAM就是旋转出来的两个角了。 + +![2024-5-30a](/images/geogebra/geogebra-2024-5-30a.png '图1辅助线') + +现在问题就变化为证明角EDG等于角FAM。 + +由三角形外角和等于不相邻两个内角的和,有 + +$\angle{AFM} = \angle{C} + \angle{CAF}$ + +另一个角DEG较复杂些,需要用到三角形内角和定理和平角定义 + +$$ +\angle{DEG} = \pi - \angle{CEG} - \angle{AED} \text{--三角形内角和等于180} \newline += \pi - (\frac{\pi}{2} - \angle{C}) - (\frac{\pi}{2} - \angle{CAF}) \newline += \angle{C} + \angle{CAF} +$$ + +显然角AFM和角DEG相等,也推出角EDG和角FAM相等。此时这两个直角三角形相似。根据三角形相似性有 + +$\frac{AF}{DE}=\frac{AM}{DG}$ + +于是问题转为求解 + +$\frac{AM}{DG}=?$ + +由题意,根据毕达哥拉斯定理很容易得到AM=4,而DG=DP+PG,且根据题意有BD=AE,由三角形相似性且为直角三角形且有斜边相等,即三角形APD与三角形ENC全等。 + +DG=DP+PG=NC+MN=MC=3,所以有 + +$\frac{AF}{DE}=\frac{AM}{DG}=\frac{4}{3}$ + +从几何原理上看,本题涉及到所谓的第一余弦定理,三角形ABC三条边为a,b,c,三边对应的角分别为A,B,C,则有。 + +$$ +a = bcos\angle{C} + ccos\angle{B} \newline +b = acos\angle{C} + ccos\angle{A} \newline +c = acos\angle{B} + bcos\angle{A} \newline +$$ + +其具有轮换对称形式,在本题中动点D、E所带动的其他点形成的轨迹构成了两个等角的直角三角形,思路也是来源于此。 + +在没有刻度尺的几何平面问题中,若想讨论线段的数量关系,第一反应就是将这些线段放到三角形中去思考,这样一来,三角形带着数量关系的定理等着我们去运用,有 +- 毕达哥拉斯定理,即勾股定理 +- 角平分线定理 +- 正弦定理 +- 余弦定理 +- 射影定理 +- 第一余弦定理 + +
+ ## 谜题类型 -### 2024-5-17 +
+2024-5-17 + 设“可怕新冠”为4位数,“新冠不可怕”为5位数,“进”,“退”为不同的一位素数,相同汉字代表相同的数字,求解谜题: $\overline{\text{可怕新冠}} \div \text{退} \times \text{进}!! \times \text{进}!! \times \text{进}!! = \overline{\text{新冠不可怕}}$ @@ -49,3 +119,5 @@ q=7\text{时}, (100x+y) \div 7!! \times (5!!)^3 = (1000y + 100t + x) \implies 22 \implies 83x = 25y \implies x = 25, y = 83. \newline \text{最后的解就是} 2583 \div 7!! \times 5!! \times 5!! \times 5!! = 83025 $$ + +
\ No newline at end of file diff --git a/exercises/quadratic.equation.md b/exercises/quadratic.equation.md index fb0145d..daa2bba 100644 --- a/exercises/quadratic.equation.md +++ b/exercises/quadratic.equation.md @@ -9,7 +9,9 @@ $$ ax^2+bx+c=0, \text{,其中 a、b、和c是已知数,而x是未知数。当这种方程的系数a、b、和c是变量而不是具体的数值时,我们称之为不定方程。} $$ -### 例题1 2024-5-16 +
+例题1 2024-5-16 + $\text{已知A、n为正整数,满足}, A = (n-7)(n+8), \text{且A为完全平方数,那么n有最几个值,n的最小值是多少,n的最大值是多少?}$ 分析:因为n在变化,可以把A缩放在某个区间来解题。使用换元法替换一个变量,这样就更容易理解了。 @@ -35,7 +37,10 @@ $$ 综上所述,n有4个值,最小的是8,最大的是56. -### 例题2 2024-5-16 +
+ +
+例题2 2024-5-16 $\text{已知} n^3 + 2n^2 + 8n - 5 \text{是一个正整数的立方,则正整数的n的值可能是}$ @@ -52,3 +57,4 @@ $$ \therefore 2^3 + 2 \cdot 2^2 + 8 \cdot 2 - 5 = 27 = 3^3 \newline \therefore 3^3 + 2 \cdot 3^2 + 8 \cdot 3 - 5 = 64 = 4^3 $$ +
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