Applying for a mathematics PhD in the US after getting a BSc degree from Imperial.

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This summer I finished my BSc maths degree at Imperial and ranked around top 5% of the class.

I was considering the idea of applying to some US universities like Stanford, Harvard, etc., but they don't do master's applications, just straight up PhD ones for mathematics. I quite like the idea of doing a PhD, so it is not a big problem for me, but I was wondering about my chances here, coming off a BSc and applying straight to a PhD sounds like a big reach.

Has anybody had similar experiences and know if it's even worth trying?

Thanks for all the help.

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