# Quod Erat Demonstrandum

## 2010/09/29

### 0 還是 1

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 11:21 上午

$(1613673)^3 + (-2262756)^3 + (1947250)^3 = ?$

## 2010/09/28

### 唔用 MI

Filed under: Pure Mathematics — johnmayhk @ 12:36 下午

Prove that

1. Show that the product of 4 consecutive positive integers is divisible by $4!$.

2. Show that $1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{n}} > 2(\sqrt{n + 1} - 1)$ ($\forall n \in \mathbb{N}$).

## 2010/09/27

### series by wolfram alpha

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 5:40 下午

Expand $\sin^{-1}x$ about $x = 0$.

http://www.wolframalpha.com/

## 2010/09/24

### 簡單的初中直線題目

Filed under: Junior Form Mathematics — johnmayhk @ 1:03 下午

1. Given that $A(4,-9), B(-4,-3), C(5,9)$, find the coordinates of the

(a) centroid（形心）;
(b) orthocenter（垂心） and
(c) circumcenter（外心） of $\Delta ABC$.

## 2010/09/23

### 唔打：基本功之節節相消

Filed under: Pure Mathematics — johnmayhk @ 3:23 下午

## 2010/09/18

### [Gag] fb

Filed under: Fun — johnmayhk @ 8:38 上午

## 2010/09/14

### 唔打：錯在哪裡之方程組

Filed under: Pure Mathematics — johnmayhk @ 8:59 下午

## 2010/09/13

### [學具][YouTube] MythBusters- Folding Paper Seven plus times

Filed under: Fun — johnmayhk @ 1:56 下午

## 2010/09/11

### [學具] logic

Filed under: Fun — johnmayhk @ 10:08 下午

## 2010/09/03

### 因式分解 x^n – 1

Filed under: Pure Mathematics — johnmayhk @ 1:06 下午

$x^2 - 1 \equiv (x - 1)(x + 1)$
$x^3 - 1 \equiv (x - 1)(x^2 + x + 1)$
$x^4 - 1 \equiv (x - 1)(x + 1)(x^2 + 1)$
$x^5 - 1 \equiv (x - 1)(x^4 + x^3 + x^2 + x + 1)$
$x^6 - 1 \equiv (x - 1)(x + 1)(x^2 - x + 1)(x^2 + x + 1)$

$\dots$

1938 年，柴巴塔洛夫（Tschebotarev）猜想：對於任何正整數 n，$x^n - 1$ 的因式（factor）不存在大於 1 或少於 -1 的係數（coefficients）。這個對嗎？ (more…)