What is the ruling on a husband and wife competing in sexual intercourse between themselves without a prize, and what is the reason for its impermissibility even if it is with the consent of both parties, and to what extent does it affect sexual arousal and the elimination of frigidity?

A competition without a prize is permissible in everything that does not involve something forbidden or distract from an obligation.

As for a competition with a prize, it is not permissible except in matters that aid in Jihad (such as camels, horses, arrows, and their equivalents among war instruments). Other than that (like football and boxing) is not permissible if it involves prizes.

The competition mentioned in the question is not permissible because it involves a prize, which is the winner's control over the issue of intercourse, and this may lead to the neglect of the non-winner's right.