# 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…)

## 2016/07/10

### 論商餘（三）

Filed under: mathematics,NSS,Teaching — johnmayhk @ 9:45 下午
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(a) 多項式與數字

## 2015/08/27

### 某 monic 多項式

Filed under: NSS,Pure Mathematics — johnmayhk @ 10:09 上午
Tags: , ,

## 2015/08/16

### 某舊習題

Filed under: Additional / Applied Mathematics,Pure Mathematics — johnmayhk @ 4:25 下午
Tags:

$a+b+c=0$

$\frac{a^5+b^5+c^5}{5}=\frac{a^3+b^3+c^3}{3}\cdot \frac{a^2+b^2+c^2}{2}$

## 2014/05/16

### 證明某級數

Filed under: Fun,Pure Mathematics — johnmayhk @ 11:00 下午
Tags: , ,

## 2011/03/03

### Polynomial identity

Is the following an identity? Prove or disprove your claim:

$(x+1)(x-3)+1=(x+2)(x-1)$

This is a trivial (more…)

## 2009/12/11

### 平移與求導

$f(x) = ax^4 + bx^3 + cx^2 + dx + e$

$f(x + L)$
$\equiv ax^4 + (4La + b)x^3 + (6L^2a + 3Lb + c)x^2$
$+ (4L^3a + 3L^2b + 2Lc + d)x + (L^4a + L^3b + L^2c + Ld + e)$ (more…)

## 2008/11/02

### 不能秒殺的提問：餘式定理

Filed under: HKCEE — johnmayhk @ 4:53 下午
Tags: ,

$f(x) \div (x - 1)$，餘式為 3
$f(x) \div (x - 2)$，餘式為 5

(more…)

## 2008/09/24

### 比較係數

Filed under: Pure Mathematics — johnmayhk @ 10:05 下午
Tags:

$f(x) = ax^2 + bx + c$$a \ne 0$），滿足 $f(f(x)) \equiv f^2(x)$，求 $a, b, c$ 的值。 (more…)