-  Climbing Mount Fuji. Articles are not used before the name of mountain.

The given sentence is grammatically incorrect because of the use of the definite article "the" before "Mount Fuji". Here are some key points to consider:

• Mount Fuji is a proper noun, which means it is the name of a specific mountain. When we use a proper noun, we do not use an article with it unless it is part of the name itself (e.g. "The United States").
• In this case, "Mount Fuji" is the name of the mountain, so we should not use the definite article "the" before it. We should simply say "Climbing Mount Fuji".
• Using "the" before "Mount Fuji" implies that there are multiple Mount Fujis and the speaker is referring to a specific one. However, this is not the case. There is only one Mount Fuji, so we should not use the definite article.

Formula:

Article usage with Proper Nouns:

• Use the definite article "the" with a singular or plural noun when it is specific or particular.
• Do not use any article before a proper noun, unless it is part of the name itself.

Example:

• The Statue of Liberty (specific)
• The Eiffel Tower (specific)
• Mount Everest (not specific)

In conclusion, the correct sentence is "Climbing Mount Fuji in winter can be very dangerous."