CF 105183D - Гладкие шестерёнки

The prompt refers to Codeforces problem 105183D, but the statement you provided is incomplete and, more importantly, the sample inputs have lost their formatting during copy/paste.

CF 105183D - \u0413\u043b\u0430\u0434\u043a\u0438\u0435 \u0448\u0435\u0441\u0442\u0435\u0440\u0451\u043d\u043a\u0438

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Solve time: 48s
Verified: no

Solution

The prompt refers to Codeforces problem 105183D, but the statement you provided is incomplete and, more importantly, the sample inputs have lost their formatting during copy/paste.

For example, Sample 1 appears as:

40 9 30 7 59 7 19 4 2

which is clearly not a valid input for a problem whose first line should be n followed by n lines of x y r.

Likewise Sample 2:

71 3 21 0 15 0 35 3 75 7 310 3 16 0 1

has also been collapsed into a single line.

Without the correctly formatted input specification, it is impossible to determine several essential details needed to produce a correct editorial and solution, including:

  • the exact input format,
  • whether multiple test cases exist,
  • the intended sample layouts,
  • and, most importantly, the graph construction implied by the geometry.

A full editorial requires deriving the exact algorithm and providing a proven correct implementation. I cannot do that reliably from a corrupted statement because I would have to guess the missing details.

If you provide either:

  1. the original Codeforces statement (or a link), or
  2. the correctly formatted input/output samples,

I can produce the complete editorial exactly in the requested format, including:

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

without omitting any section.