# Quod Erat Demonstrandum

## 2021/01/19

### 無聊談對稱多項式

Filed under: Junior Form Mathematics,mathematics,NSS — johnmayhk @ 11:51 上午
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$(a+b+c)^5-a^5-b^5-c^5$

$f(a)=(a+b+c)^5-a^5-b^5-c^5$

(more…)

## 2021/01/01

### 又談無聊不等式

Filed under: mathematics,NSS,Pure Mathematics — johnmayhk @ 1:10 下午
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$2^n > 6n^2+8n+9$ for all integers $n \ge 10$

?

(more…)

## 2020/12/28

### 無聊談兩個不等式

$a+b > c$$b+c > a$　及　$c+a > b$

$2(a^2+b^2) > c^2$

(more…)

## 2020/05/21

### Basic question of differentiation

Filed under: Additional / Applied Mathematics,mathematics,NSS — johnmayhk @ 7:20 下午
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HKDSE 2020 M2 Q.9 (b), a 2-mark question:

Given $\displaystyle f(x)=\frac{(x+4)^3}{(x-4)^2}$, find $f''(x)$.

How fast can you finish this part and obtain the correct answer, especially when you are under the pressure during the public examination?

3 minutes? (2/100 * total time allowed = 2/100 * 150 minutes)

(more…)

### Similar-looking formula

Filed under: Junior Form Mathematics,mathematics,Physics — johnmayhk @ 4:01 下午
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The equivalent resistance $R$ of a parallel circuit

can be determined by

$\displaystyle \frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$.

A similar-looking formula found in a basic mathematics question involving parallel lines as shown below:

## 2020/02/28

### 正弦積

$\tan 1^o\tan 2^o\tan 3^o\dots \tan 88^o\tan 89^o$

$\tan \theta \tan (90^o-\theta) \equiv 1$

$\tan 1^o\tan 2^o\tan 3^o\dots \tan 88^o\tan 89^o$
$=(\tan 1^o\tan 89^o)(\tan 2^o\tan 88^o)\dots (\tan 44^o\tan 46^o)\tan 45^o$
$=1\times 1\times \dots \times 1$
$=1$

$\sin 1^o\sin 2^o\sin 3^o\dots \sin 88^o\sin 89^o$

(more…)

## 2019/12/13

### 受保護的文章：F.4 Core Math Quiz Ch.2

Filed under: mathematics,NSS — johnmayhk @ 10:16 上午
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## 2019/12/08

### 受保護的文章：F5 M2 RT 20191206

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 11:49 上午
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## 2019/05/10

### 線長乘積

Filed under: Pure Mathematics — johnmayhk @ 11:52 下午
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## 2019/05/05

### What’s wrong?

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 7:05 下午
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Here is a basic level M2 question:

Given that $\sqrt{xy}=7+2y$, find $\frac{dy}{dx}$ at ($-\frac{1}{3}$,$-3$).

Student 1 gave

$\frac{1}{2\sqrt{xy}}(x\frac{dy}{dx}+y)=2\frac{dy}{dx}$

$\frac{1}{2}\sqrt{\frac{x}{y}}\frac{dy}{dx}+\frac{1}{2}\sqrt{\frac{y}{x}}=2\frac{dy}{dx}$

$\frac{dy}{dx}=\sqrt{\frac{y}{x}}\cdot\frac{1}{4-\sqrt{\frac{x}{y}}}$

Thus, at ($-\frac{1}{3}$,$-3$),

$\frac{dy}{dx}=\sqrt{\frac{-3}{-1/3}}\cdot\frac{1}{4-\sqrt{\frac{-1/3}{-3}}}=\frac{9}{11}$

Student 2 gave (more…)

## 2019/02/01

### 帕斯卡三角某結果

Filed under: NSS,Pure Mathematics — johnmayhk @ 5:35 下午
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$f^{[1]}(x)=f(x)$

$f^{[2]}(x)=f(f(x))$ (more…)

## 2018/11/20

### 費氏講

Filed under: Additional / Applied Mathematics,Fun,Junior Form Mathematics — johnmayhk @ 6:36 下午
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N 年前往中一班代堂，必談「64 = 65」謎題：

(圖片來源：https://i.stack.imgur.com/fWdMd.jpg)

## 2018/11/02

### 受保護的文章：SFXC F.3A Mathematics Assignment 11 (Web Task)

Filed under: Teaching — johnmayhk @ 9:19 上午

## 2018/09/16

### 長周素

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 7:20 下午
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$\frac{1}{7}=0.\overline{142857}$ ，故循環周期（decimal period）為 6。

$\frac{1}{17}=0.\overline{088235294117647}$ ，循環周期為 16。

$\frac{1}{19}=0.\overline{052631578947368421}$，循環周期為 18。

## 2018/09/15

### 用積分證 0=-1

Filed under: NSS — johnmayhk @ 6:01 下午
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$\int \tan xdx=\int \frac{\sin xdx}{\cos x}=-\int \frac{d\cos x}{\cos x}=-\frac{\cos x}{\cos x}+\int \cos xd(\sec x)=-1+\int \tan xdx$

$0=-1$ (more…)