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You probably get Euler's constant $\gamma$ when you do the integral test comparing $\sum\frac1n$ to $\int\frac{dx}{x}$. Then you can remark that it is unknown whether $\gamma$ is rational.


It's still not known whether $$\zeta(5) = \sum_{n=1}^\infty \frac{1}{n^5}$$ is a rational number.


This is a bit obvious I think, but when you introduce sequences and their notation in either an algebra or calculus class, you should certainly show students the Collatz Conjecture as one of the examples.


It takes a lot of browsing to find problems somehow related to calculus or analysis, but this is a great MathOverflow list: Not especially famous, long-open problems which anyone can understand. Here are a few from that list: Are there an infinite number of primes $p$ such that the repeating part of the decimal expansion of $1/p$ has length $p-1$? Link. ...


[VERY LONG ANSWER, needs patience to read through] I feel this is a problem many students who are good at maths face. They understand the simple tricks and patterns which are present in the school syllabus and so it is simple for them and after some practise and memorisation they are done. Then they seek out more maths and find out about topics like ...


I will also chime in and say that the argument on the linked Art of Problem Solving site is unpersuasive, and somewhat misses a broader point. Ultimately, the real point of the mathematical discipline is to identify patterns in systems and prove their correctness (hopefully in an insightful, persuasive, explanatory style). The "trap" that I would identify ...


The article at artofproblemsolving seems silly to me. The author's idiosyncratic opinion seems to be that students who are ready to take calculus should refrain from taking calculus and instead do math contests. People are all different, and there is not just one appropriate path for a mathematically precocious student. Some people might want to take ...


Echoing @AndreasBlass' remark, and having experienced somewhat similar episodes, it is already precarious enough to make such choices _for_oneself_. So, to directly answer your question: I think "no, do not encourage others to (too violently) disconnect from the math curriculum at school". I don't think it's about problem-solving versus calculus, at all. And,...

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