Is it permissible to pay a bribe to get a job for which I am qualified, knowing that it is not possible to obtain the job without paying this bribe?

If a Muslim is subjected to injustice, or is entitled to a right, and cannot repel the injustice or obtain his right except by paying money, then he may do so. He is not considered a briber, but rather this falls under the category of 'mudaraah' (conciliatory behavior) and averting harm, and the sin rests upon the one who takes [the money]. It is narrated that Ibn Mas'ud gave two dinars to be released in Abyssinia, and it is narrated from the Tabi'in (successors) that there is no harm in a man conciliating (using 'musana'ah') for himself and his money if he fears injustice. Therefore, if you are certain that your qualifications entitle you to the job, and that you will perform the work perfectly, then there is no harm in paying money to obtain it, especially if the job cannot be obtained otherwise.