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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, ### Donate to Tolaso Network 