3203. The Balance

单点时限: 5.0 sec

内存限制: 256 MB

Ms. Iyo Kiffa-Australis has a balance and only two kinds of weights to measure a dose of medicine. For example, to measure $200\mathrm{mg}$ of aspirin using $300\mathrm{mg}$ weights and $700\mathrm{mg}$ weights, she can put one $700\mathrm{mg}$ weight on the side of the medicine and three $300\mathrm{mg}$ weights on the opposite side (Figure 1). Although she could put four $300\mathrm{mg}$ weights on the medicine side and two $700\mathrm{mg}$ weights on the other (Figure 2), she would not choose this solution because it is less convenient to use more weights.

You are asked to help her by calculating how many weights are required.


The input is a sequence of datasets. A dataset is a line containing three positive integers $a$, $b$, and $d$ separated by a space. The following relations hold: $a \neq b$, $a \leq 10000$, $b \leq 10000$, and $d \leq 50000$. You may assume that it is possible to measure $d$ mg using a combination of $a$ mg and $b$ mg weights. In other words, you need not consider no solution cases.

The end of the input is indicated by a line containing three zeros separated by a space. It is not a dataset.


The output should be composed of lines, each corresponding to an input dataset $(a, b, d)$. An output line should contain two nonnegative integers $x$ and $y$ separated by a space. They should satisfy the following three conditions.

  • You can measure $d\mathrm{mg}$ using $x$ many $a\mathrm{mg}$ weights and $y$ many $b\mathrm{mg}$ weights.
  • The total number of weights $(x + y)$ is the smallest among those pairs of nonnegative integers satisfying the previous condition.
  • The total mass of weights $(ax + by)$ is the smallest among those pairs of nonnegative integers satisfying the previous two conditions.

No extra characters (e.g. extra spaces) should appear in the output.


700 300 200
500 200 300
500 200 500
275 110 330
275 110 385
648 375 4002
3 1 10000
0 0 0
1 3
1 1
1 0
0 3
1 1
49 74
3333 1

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创建: 3 年,5 月前.

修改: 2 年,11 月前.

最后提交: 2 年,5 月前.

来源: Japan 2004