Determine the Price（ZJU 2002）
Determine the Price
时间限制: 1 Sec 内存限制: 128 MB
题目描述
For the manager of a theatre, setting the price of a ticket is a rather delicate matter. Suppose that a theatre has n (<= 1000000) seats, and that if you give away the tickets for free, all the seats will be taken. After some sort of investigation, it is estimated that for every p yuans of increase of the ticket price, the theatre will lose m audiences. But the audiences will not leave until the ticket price jumps up a complete step of p yuans. For example, if given p=1.00 and m=10, no audience will be lost if the price is set to be 0.99 yuans, and 10 will be lost if the price is 1.99 yuans. It costs R yuans for the theatre to run each show, and S yuans to serve each of the audiences during each show. Now your job is to write a program to maximize the total revenue of each show for the theatre.
输入
The first line of input contains a positive integer N (< 10000), then followed by N test cases.
Each test case consists of 5 positive numbers in a line, n, p, m, R, S which are all defined in the problem.
输出
For each test case, print the optimal price of a ticket, the total number of audiences to sit in the theatre with that price set, and the maximum revenue, in the format illustrated by the sample output.
Note: the unit of price is RMB yuan, and a price must be rounded off to 2 decimal places. There must be one blank line between two test cases, but no extra line at the end of output.
样例输入
复制样例数据
2 1000 0.10 10 1000 2.00 100 0.50 2 500 2.0
样例输出
price = 6.09 audiences = 400 revenue = 636.00 price = 13.99 audiences = 46 revenue = 51.54
数学解法： 我们可以很轻易的得到这样一个二次函数。
Y = PK*（N – MK）- R -（N- MK）S
Y表示收益， K表示票价上升的倍数。 则整理得：
Y= -PMK^2 + (PN + MS)S – NS – R
二次函数的顶点坐标为 ( - b / 2a , (4ac-b^2)/4a ) ,
此函数的顶点X坐标为 – b / 2a = - (PN + MS) / 2*(-PM) = (PN + MS) / 2PM
得到这个值，则为最优值，但不能保证精确（因为毕竟答案只取2位小数），所以我们加以部分枚举， 枚举 X – 10 ~ X + 10， 这样就可以保证答案的正确性了。
得到了X，可以计算各个值了：
Price = P * X – 0.01
Audiences = N - M * ( X - 1 )
Revenue = Price * Audiences - R - S * Audiences
/**/
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#include 好久没写博客了，有很多数论的题目没总结呢，与其说是没时间，还不如说是懒得写，哎。。