Evaluation of hopeful candidate (Oxford, but also Stanford, MIT, etc.)

Hi ladies and gents!

I'm sorry in advance if this is not the right forum/place for these types of questions; I'm new here and thought I'd just throw this out there anyway.

I am currently a second-year student who plans on applying to Oxford's Mathematics and Foundations of Computer Science Master's programme next year. Now, I could not find much information about the other applicants and their qualifications (or, well, a mean-field approximation of this), hence this thread/questions. It'd be great to hear whether you guys think my profile would be competitive enough, and if you have any examples of "profiles" that got admitted or rejected. I also plan on applying to PhD programmes in the US, such as Stanford, MIT, CMU and so forth (as well as perhaps a master's programme at UCL as well, serving as back-up).

About me:

I'm doing a double major in math/CS at a top 100 uni in Europe, with a GPA of about 82/100, which is just above Oxford's requirement. I am enrolled in the honours programme and have also been somewhat successful in competitions in math/CS. For instance, I got top 20 in a national math competition for uni students, which is the uni-analogue of high school olympiads. I also got some medals at the national level for the olympiads, although I'm not sure whether this would be worth much for grad applications.

Furthermore, my uni has a lot of "research" opportunities, and I have taken advantage of that as well, although it'd be a huge stretch to say it'd be anywhere near worth publishing in any self-respecting paper. However, I think that taking these opportunities and being enrolled in courses in which one is allowed to carry out research is actually valuable on its own; do you think I should mention it in my letter of motivation (having extensive research experience), or just let it be, since there's nothing significant?

I'm curious to see what you guys think! :-)