1. Time Machine 時光倒流

Saw an interesting ad on the latest Adobe product and said, "With the new product runs 50% faster, meaning save you 50% of time".  It is interesting to think if the product runs 150% faster, do we save 150% of time?  If a job need to take 1 day to finish, 150% of time less will take us back approximately yesterday noon.

曾看過某軟件的廣告謂：『新產品的運作比從前快百分之五十，從此節省閣下百分之五十的時間。』想想看，假若新軟件的運作快了百分之一百五十，時間會否也節省了百分之一百五十？假若工序需時一天，節省了百分之一百五十的時間，大既會回到昨天中午左右。

Solution

2. 1=-1

Remark: This one similar to the 2=1 problem of last part.

註意：這題跟上次2=1問題雷同

Solution

3. a>b and a=b?

let a>b and therefore c>0, and

       a=b+c
  a(a-b)=(b+c)(a-b)
   a²-ab=ab+ac-b²-cb
a²-ab-ac=ab-b²-cb
a(a-b-c)=b(a-b-c)
       a=b

Solution

4. （n-2)²=n²? (require Linear Algebra)

Prove or disprove (n-2)²=n²

S_n: (n-2)²=n²

Let n=k, and k+1 imply

S_k+1:     (k-1)²=(k+1)²
          k²-2k+1=k²+2k+1
            -2k=2k
         -2k-2k=0
            -4k=0
     -4k(1-1/k)=0(1-1/k)
          -4k+4=0
        k²-4k+4=k²
           (k-2)²=k²

Since the equation is true for n and n+1, therefore we can conclude that (n-2)=n² for all n>0.

Solution

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Wilson
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