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# Tag Archives: Contests

## Rational function and polynomial

Prove that there does not exist a rational function with real coefficients such that

where is a non constant polynomial.

**Solution**

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.

## Sum over all positive rationals

For a rational number that equals in lowest terms , let . Prove that:

**Solution**

First of all we note that

Moreover for we have that

Hence for we have that

## Infinite Product

Evaluate the infinite product:

*(IMC 2019 / Day 1 / Problem 1)*

**Solution**

since the product telescopes.

## On an infinite summation

Let be a sequence of real numbers. Compute:

**Solution**

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,

## Trigonometric equality

Prove that in any triangle it holds that

where denotes the circumradius and the inradius.

**Solution**

Using the law of sines we have that

and if we denote the area of the triangle then

Thus,