CF 105174G - 溶液配制 Ⅱ

I don't have enough information to write a correct editorial or produce a correct solution for this problem. The statement you've provided is corrupted in a crucial place.

codeforcescompetitive-programming

CF 105174G - \u6eb6\u6db2\u914d\u5236 \u2161

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Solve time: 31s
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Solution

I don't have enough information to write a correct editorial or produce a correct solution for this problem.

The statement you've provided is corrupted in a crucial place. The mathematical condition defining when a subset can produce the target concentration is missing because the formulas were mangled during extraction. The relevant part currently appears as:

存在非负实数 $p_1,\dots,p_m$，使得

...

$= x$，

...

$= y$

but the actual equations involving the weighted averages are lost.

Those missing equations completely determine the mathematics of the problem. For example, the intended condition is likely something equivalent to

$$\frac{\sum p_i v_{a_i}}{\sum p_i}=x,\qquad \frac{\sum p_i w_{a_i}}{\sum p_i}=y,$$

but I cannot safely assume that from the damaged text. A competitive programming editorial, proof, algorithm, and implementation must exactly match the official problem, and even a small difference in these equations changes the entire solution.

Please provide one of the following:

  1. The original Codeforces problem link for 105174G - 溶液配制 II, or
  2. The PDF page with the formulas visible, or
  3. A clean copy of the statement containing the mathematical equations.

Once I have the complete statement, I'll produce the full editorial in the exact format you requested, including:

  • Problem Understanding
  • Approaches
  • Algorithm Walkthrough
  • Correctness proof
  • Complete Python solution
  • Worked examples
  • Complexity analysis
  • Assert-based test cases
  • Edge case analysis