# Quod Erat Demonstrandum

## 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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## 2018/07/18

### 畫 y=x^(1/n)

Filed under: Additional / Applied Mathematics,mathematics,NSS — johnmayhk @ 12:57 下午
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$\sqrt[3]{4}-\sqrt[3]{3}$$\sqrt[3]{3}-\sqrt[3]{2}$

## 2018/07/13

### pi的連分表達式

Filed under: mathematics,NSS — johnmayhk @ 2:52 下午
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$\pi$ 寫成連分數（continued fraction），方式甚多，以下是其中一種：

Define $\displaystyle I_n=\int_0^1\frac{x^{2n}}{1+x^2}dx$. (more…)

## 2018/03/22

### a question about inequality with derivatives

Filed under: Fun,mathematics,NSS,Pure Mathematics — johnmayhk @ 3:46 下午
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Question

Let $p(x)$ be a polynomial with real coefficients. Prove that if $p(x)-p'(x)-p''(x)+p'''(x)\ge 0$ for any real $x$, then $p(x) \ge 0$ for any real $x$.

Solution (elementary) (more…)

## 2018/03/14

### 黃金比某級數

Filed under: Fun,mathematics,NSS — johnmayhk @ 11:11 上午
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$\displaystyle \Phi=\frac{1}{\Phi}+\frac{1}{\Phi^2}+\frac{1}{\Phi^3}+\dots$

$\displaystyle \Phi=\frac{1+\sqrt{5}}{2}$

(more…)

## 2018/02/22

### 懷古-開方2

Filed under: Fun,mathematics — johnmayhk @ 6:08 下午
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$\displaystyle \sqrt{2} \simeq 1+\frac{1}{3}+\frac{1}{3\times 4}-\frac{1}{3\times 4\times 34}$

## 2018/01/23

### 某類三角恆等式記法

Filed under: mathematics,NSS — johnmayhk @ 10:24 下午
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$\sin(-\theta)\equiv -\sin\theta$
$\cos(-\theta)\equiv \cos\theta$
$\tan(-\theta)\equiv -\tan\theta$

## 2017/11/14

### as gs

Filed under: Additional / Applied Mathematics,mathematics,NSS — johnmayhk @ 12:29 上午
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Derive the formula for the sum of first $n$ terms of the following sequence in terms of $a,b,d,r,n$, where $r \ne 1$.

$ab,(a+d)br,(a+2d)br^2,(a+3d)br^3,\dots$

## 2017/10/28

### 一題多解

Filed under: Junior Form Mathematics,mathematics,NSS — johnmayhk @ 12:14 上午
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（不過有多少學生解完題目，會如此神心尋求另外解法？面對極度規範化的考題，方法多數固定，對一些同學來說，莫說一題多解，更多時是找不到解法。）

## 2017/09/21

### core math 某題：標準差

Filed under: mathematics,NSS — johnmayhk @ 4:49 下午
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http://www.wolframalpha.com/input/?i=standard+deviation+of+1,1,1,2,2,2,2,2,3,3,4,4,4,5

## 2017/08/12

### 重積求面積

Filed under: mathematics,NSS,University Mathematics — johnmayhk @ 6:10 下午
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1.某中四題：

## 2017/06/23

### 實數問題複數解決

Filed under: mathematics,NSS — johnmayhk @ 3:43 下午
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$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$

## 2017/06/21

### 兩題二次方程

Filed under: mathematics,NSS — johnmayhk @ 6:28 下午
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1.

Refer to the figure below.

If $\alpha$ and $\beta$ are x-coordinates of P and Q respectively such that $\alpha^2+\beta^2=13$, find the value(s) of m.

$-x^2+3x-2=mx-8$
$x^2+(m-3)x-6=0$ (more…)

## 2017/06/10

### 正多邊形方程

Filed under: Additional / Applied Mathematics,Fun,mathematics,NSS — johnmayhk @ 12:24 下午
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https://www.desmos.com/calculator/vv7stc4nl0