puzzleup 2016

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In[47]:= x

Out[47]= {2, 3, 4, 6, 9, 12}
In[3]:= y=Subsets[x,{4}]
In[48]:= Exp/@Total/@Log/@y / Total[x]

Out[48]= {4, 6, 8, 9, 12, 18, 12, 16, 24, 36, 18, 24, 36, 54, 72}

〇〇

刚才写错了,36不是已知的,e>d而不是e>f
In[2]:= Solve[{s=a+b+c+d+e+f,Mod[abcd,s]==0&&Mod[abce,s]==0&&Mod[abcf,s]==0&&Mod[abde,s]==0&&Mod[abdf,s]==0&&Mod[acde,s]==0&&a>=1&&b>a&&c>b&&d>c&&e>d&&f>e},{a,b,c,d,e,f}, Integers]

Solve::eqf: a >= 1 is not a well-formed equation.

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〇〇 发表于 2016-9-11 18:22
In[47]:= x

Out[47]= {2, 3, 4, 6, 9, 12}


找到了mma简便的求积写法
In[4]:= n={2,3,4,7}

Out[4]= {2, 3, 4, 7}

In[5]:= n[[1]]

Out[5]= 2
In[6]:= Product[n[],{i,1,4}]

Out[6]= 168

In[7]:= x= {2, 3, 4, 6, 9, 12}

Out[7]= {2, 3, 4, 6, 9, 12}

In[8]:=  y=Subsets[x,{4}]

Out[8]= {{2, 3, 4, 6}, {2, 3, 4, 9}, {2, 3, 4, 12}, {2, 3, 6, 9}, {2, 3, 6, 12}, {2, 3, 9, 12},

>    {2, 4, 6, 9}, {2, 4, 6, 12}, {2, 4, 9, 12}, {2, 6, 9, 12}, {3, 4, 6, 9}, {3, 4, 6, 12},

>    {3, 4, 9, 12}, {3, 6, 9, 12}, {4, 6, 9, 12}}

In[9]:= Plus@@@y

Out[9]= {15, 18, 21, 20, 23, 26, 21, 24, 27, 29, 22, 25, 28, 30, 31}

In[10]:= Times@@@y

Out[10]= {144, 216, 288, 324, 432, 648, 432, 576, 864, 1296, 648, 864, 1296, 1944, 2592}

〇〇

No: 08       September 14, 2016  



BALLOONS

BALLOONS

In the play room there are two identical boxes each having 10 balloons. A boy randomly (with equal probability) selects a box and bursts a balloon. He continues this process until the box he selected turns out to be empty. When he stops what is the probability that the other box has exactly 5 balloons?

Enter your answer as a simplified fraction. Example:12/23

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2个同样的盒子里各有10个气球,每次抽1个盒子,刺破1个球,继续这么做,直到盒子里为空停止,当他停止时,另一个盒子里正好5球的概率?

newkid

奥数题的水平，如果用SQL全部模拟出来未尝不可，总共才(2^20)种情况。

lugionline

用sql不模拟可以算出精确比例

上题81以后的下一个解并非256，所以理解还是不对
假设 sum 有一个因子 p^k,那么每个色子必有因子 p^[k/4] 这里[ ] 表示Ceiling
于是 sum = p1^k1 * p2^k2 * ... * pr^kr
               = p1^[k1/4] * p2^[k2/4] * ... * pr^[kr/4] * (a + b + c + d + e + f)
其中 a + b + c + d + e + f >= 1 + 2 + 3 + 4 + 5 + 6 = 21
于是 a + b + c + d + e + f = Sigma[pi^(ki - [ki/4])] >= 21 时必有解
下一组解是：
{dict -> {5, 10, 20, 25, 30, 35}, sum -> 125}

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lugionline 发表于 2016-9-15 06:56
用sql不模拟可以算出精确比例

上题81以后的下一个解并非256，所以理解还是不对

看懂了

newkid

lugionline 发表于 2016-9-15 06:56
用sql不模拟可以算出精确比例

上题81以后的下一个解并非256，所以理解还是不对

你来写个SQL看看？
如果穷尽十次的所有组合，思路很简单代码也很少，当然效率就不怎么样。用排列组合公式我只想到比较复杂的做法，不知道有什么巧妙的办法。

lugionline

这个对吗，不写SQL，否则又是一大堆



PS.我用通项公式验证了下，貌似正确