Quod Erat Demonstrandum

2017/03/20

無聊 bonus

Filed under: mathematics,NSS — johnmayhk @ 9:55 上午
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Solve the following equations for real $x$.

1. $1+9^x+25^x=3^x+5^x+15^x$

2. $5^{x+1}+5(2^x)=3(10^x)+25^x+4^x+5$

2016/12/11

小心出題

Filed under: mathematics,NSS — johnmayhk @ 11:18 下午
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$AB=\sqrt{30^2+45^2-2(30)(45)\cos(60^o-40^o)}=19.7$ m

$AB=45\cos 40^o-30\cos 60^o=19.5$ m

2016/08/08

點解2

Filed under: Fun,mathematics — johnmayhk @ 11:12 下午
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$1+2+3+\dots +(n-1)+n+(n-1)+\dots +3+2+1=?$

$\frac{n(n+1)}{2}+\frac{(n-1)n}{2}=n^2$

$1+2+3+\dots +(n-1)+n+(n-1)+\dots +3+2+1=n^2$

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

2016/06/06

和扇形有關的某積分

Filed under: mathematics,NSS — johnmayhk @ 12:27 下午
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$\displaystyle \int^1_0\sqrt{1-x^2}dx$

$\displaystyle \int^1_0\sqrt{1-x^2}dx=\frac{\pi}{4}$

$\displaystyle \int_0^t\sqrt{1-x^2}dx$　（其中 $0 < t < 1$

$x=\sin\theta$ 又或用部分積分吧。但以圖示之，求上述積分，即是求下圖著色部分面積：

(more…)

2016/04/07

log y to the base a

Filed under: Fun,mathematics — johnmayhk @ 4:11 下午

（想畫公仔，不喜勿插）

15，16 世紀某天。三條書局友，在表面寧靜的式子…

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2016/02/15

某 front view

Filed under: Junior Form Mathematics,mathematics — johnmayhk @ 2:38 下午

2015/07/07

某關於整除的題

Filed under: mathematics,NSS — johnmayhk @ 3:06 下午
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$(n^2)!$

$(n!)^{n+1}$

(more…)

2015/03/07

幾條正三角形

Filed under: mathematics,NSS — johnmayhk @ 8:59 下午
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2015/02/17

數算球入盒

Filed under: mathematics,NSS,Pure Mathematics — johnmayhk @ 11:40 上午
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（一） (more…)

2015/02/01

黑白球

Filed under: mathematics,NSS — johnmayhk @ 10:33 上午
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In a game, Evan has to draw balls from a bag containing 2 black balls and 3 white balls one by one without replacement. If he gets two consecutive black balls, he wins; otherwise he loses. Find the probability that he wins.

P(wins)
=P(BB)+P(WBB)+P(WWBB)+P(WWWBB)
=$\frac{2}{5}\frac{1}{4}+\frac{3}{5}\frac{3}{4}\frac{2}{3}+\frac{3}{5}\frac{2}{4}\frac{2}{3}\frac{1}{2}+\frac{3}{5}\frac{2}{4}\frac{1}{3}$
=$\frac{2}{5}$

$\frac{3}{7}\frac{2}{6}+\frac{4}{7}\frac{3}{6}\frac{2}{5}+\frac{4}{7}\frac{3}{6}\frac{3}{5}\frac{2}{4}+\frac{4}{7}\frac{3}{6}\frac{2}{5}\frac{3}{4}\frac{2}{3}+\frac{4}{7}\frac{3}{6}\frac{2}{5}\frac{1}{4}=\frac{3}{7}$

2015/01/23

某數算題

Filed under: mathematics,NSS — johnmayhk @ 5:34 下午
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Just reply to a F.5C student on a basic core mathematics question (on P.5.38):

There are 8 outstanding students from junior forms and 9 outstanding students from senior forms in a school this year. 5 out of these 17 students are now selected for an overseas exchange programme. Find the number of combinations of selecting at least 1 student from junior forms and 1 from senior forms.

Here is the ‘so-called’ solution from a student:

$_8C_1\times _9C_1\times _{15}C_3$

as the student claimed, select 1 from junior, $_8C_1$ ways; select 1 from senior, $_9C_1$ ways; then select the remaining 3 students from the remaining 15 students, $_{15}C_3$ ways, hence, the total number of combination should be $_8C_1\times _9C_1\times _{15}C_3$, right?

Sorry, it is incorrect. (more…)

2015/01/12

just a core math question involving variance

Filed under: mathematics,NSS — johnmayhk @ 10:35 下午

Just reply to a student about a core math question used in school exam.

Question

Let $m$, $v$ be the mean and variance of {$2x_1, 2x_2, 2x_3$} respectively.

Show that the mean and variance of the set

{$x_1+x_1,x_1+x_2,x_1+x_3,x_2+x_1,x_2+x_2,x_2+x_3,x_3+x_1,x_3+x_2,x_3+x_3$}

are $m$ and $\frac{v}{2}$ respectively.

Proof

(Mean) For the original set, (more…)

2015/01/08

被 8 整除

Filed under: Fun,mathematics — johnmayhk @ 2:08 下午
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1.問題

$n^4+2n^3+3n^2+2n$

2.解答

(a)

2014/12/02

有關數學家張益唐的報導

Filed under: mathematics,Report — johnmayhk @ 10:47 下午
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