PokéBase - Pokémon Q&A
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The score earned for a given Pokémon in Generation IV is the sum of the following:

The level of the Pokémon relative to the maximum that can be found in the contest (e.g. before the National Pokédex, the maximum value for Paras would be 17, but after the National Pokédex, this would be 34), as a percentage
The Pokémon's IVs relative to the maximum (186), as a percentage
The Pokémon's HP relative to its maximum, as a percentage
A rarity factor, either 60 (Caterpie, Metapod, Weedle, Kakuna, Wurmple, Silcoon, Cascoon, or Kricketot), 80 (all Pokémon not mentioned), or 100 (Scyther or Pinsir)
Therefore, the maximum possible score in this version of the contest is 400, achieved by catching Scyther or Pinsir at either level 14 or 28 (whichever is available), with maximum IVs in every stat, and without dealing any damage to it.

So, the factors that go into your scoring are the Pokemon IV's, HP, rarity, and level.

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catching rarer Pokemon like syther or pinsir will win you the contest. sometimes even butterfree or beedrill also win you the contest don't catch weedles and caterpies that's it.
source: shiny hunting syther and many other contests I did
Hope I Helped!