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Prove that there does not exist a rational function with real coefficients such that
where is a non constant polynomial.
Since polynomials are defined on we have that
Since tends to a finite value as it must be a constant polynomial. In particular, must be constant in the range of which is an infinite set, implying that must also be constant. This proves what we wanted.
For a rational number that equals in lowest terms , let . Prove that:
First of all we note that
Moreover for we have that
Hence for we have that
Evaluate the infinite product:
(IMC 2019 / Day 1 / Problem 1)
since the product telescopes.
Let be a sequence of real numbers. Compute:
First and foremost we set and it is obvious that . We are making use of probabilistic methods. Suppose than an infinite number of coins are flipped. Let be the probability that the -th coin toss lands heads and let us consider the first time heads comes up. Then is the probability that the first head appears in the – th flip and is the probability that all flips come up tails. Thus,
Prove that in any triangle it holds that
where denotes the circumradius and the inradius.
Using the law of sines we have that
and if we denote the area of the triangle then