Challenging JEE Questions: Impress or Horrify Your US Students?

My son is currently undergoing JEE coaching, and as a math enthusiast, I periodically monitor his studies. Recently, I came across a few problems that I found particularly interesting. Here are some challenging JEE questions that could impress or horrify your US students with a comparative perspective.

Problem 1: Infinite Product Series

Let 1 and let A_n frac{x}{1-x^2} frac{x^2}{1-x^4} cdots frac{x^{2^n}}{1-x^{2^{n-1}}}). What is lim_{n to infty} A_n)?

Problem 2: Complex Numbers and Powers

If z frac{sqrt{3}-i}{2}), then what is the smallest integer value of n) for which z^{95}i^{67} z^n)?

Note: (i) is the imaginary unit, where i sqrt{-1}).

Problem 3: Rolle's Theorem Application

Let f) and g) be two functions continuous on [a, b]) and differentiable on (a, b]). Suppose f(a) f(b) 0). Prove that there is a point c in (a, b)) such that fc cdot g'(c) - f'(c) 0).

Problem 4: Limit and Integral Representation

Let P lim_{n to infty} left( prod_{r1}^n n^3 r^3 right)^{1/n} / n^3) and lambda int_{0}^{1} frac{dx}{1 - x^3}). Express ln P) in terms of lambda).

Problem 5: Matrix Analysis

Let n geqslant 3) and A a_{ij}_{1 leqslant i, j leqslant n}) be an n times n) matrix such that a_{ij} in {-1, 1}) for all 1 leqslant i, j leqslant n). Suppose that a_{k1} 1) for all 1 leqslant k leqslant n) and sum_{k1}^n a_{ki} a_{kj} 0) for all i eq j). Show that n) is a multiple of 4.

These are some of the problems that Indian students have to solve for JEE (Joint Entrance Examination).

Note: Problem 5 is not from JEE but from the ISI (Indian Statistical Institute) UG entrance exam, which is taken by 17-year-olds, the same age group as those preparing for JEE. I would classify this problem as ultra-difficult.

These questions have impressed/horrified me for the last two years. It was incredibly difficult for me to pick just a few out of the vast expanse of such challenging problems. Here are a few questions that just skim the tip of the iceberg.

I am sure there are myriad other exceedingly challenging problems in physics and chemistry as well, but these math problems sure gave me some sleepless nights.

While these questions may seem extremely easy for those preparing for JEE advanced or other competitive exams, these problems are still miles away from the math tested in US exams such as the SAT, ACT, or even the SAT Subject Level 2.

Let's discuss your thoughts and solutions in the comments below!