Firstly: 

The Eids and special occasions which are prescribed for the Muslim ummah are few and well known, as sharee’ah has told us of them and encourages people to celebrate them. They include special occasions such as Ramadaan, the Eids, the first ten days of Dhu’l-Hijjah and Muharram, and so on. But they do not include the Prophet’s Birthday (Mawlid), because there are no texts that single this day out as a special occasion, or for any particular act of worship or celebration. Rather the Sahaabah and Taabi’een and those who came after them did not regard it as a special occasion. Those who regard it as being prescribed by sharee’ah in any sense have introduced an innovation into the religion that is not part of it. We have stated previously on this site that it is an innovation (bid’ah) to celebrate the Prophet’s Birthday. 

Please see the answers to questions no. 5219, 10070, 13810, 20889 and 70317. 

Secondly: 

Undoubtedly holding competitions on that day is a kind of celebrating it and making it an Eid, so it is not permissible to take part in any competition that is held on an innovated occasion, otherwise the one who is taking part will also be counted as a follower of innovation. We ask Allaah to keep us safe and sound. 

It says in Fataawa al-Lajnah al-Daa’imah (3/25): 

What do you think – may Allaah preserve you as a help to the Muslim ummah – about taking days off from school and work, or giving sermons and lectures etc, as is done in our country in Africa, on the occasion of the Prophet’s Birthday? 

The answer was: 

Celebrating the Prophet’s Birthday and taking a day off for that is an innovation (bid’ah), because the Prophet (peace and blessings of Allaah be upon him) did not do that and neither did his companions (may Allaah be pleased with them). The Prophet (peace and blessings of Allaah be upon him) said: “Whoever introduces anything into this matter of ours that is not part of it will have it rejected.” End quote. 

Thirdly: 

With regard to Islamically-prescribed occasions, such as the month of Ramadaan etc, what is prescribed and in fact mustahabb is to remind the people of it, and to inform them of its virtues and the deeds that are mustahabb at that time, and the rewards decreed for that; organizing lessons and seminars is the best way to teach the people how to celebrate the special occasions that are prescribed in Islam. 

Among the ways of celebrating these special occasions is holding competitions of knowledge and Qur’aan memorization during these special occasions, when the people turn to Allaah and strive to read Qur’aan, memorize it and learn the rulings of Islam. So there is nothing wrong with organizing such competitions or taking part in them, in sha Allaah. 

Fourthly: 

We have previously discussed on our site the ruling on offering prizes in different kinds of competitions. The correct view is that it is permissible if the competition brings some kind of religious or worldly benefit. The Hanafis have stated that it is permissible to offer prizes for competitions in knowledge and mathematics. 

It says in al-Fataawa al-Hindiyyah (5/324): 

If one of the educated people were to say to one of his peers: Come, let us compete and if you get it right and I get it wrong I will give you such and such, and if I get it right and you get it wrong, I will not take anything from you, it should be permissible. End quote. 

See: Radd al-Muhtaar (6/4040. 

And Allaah knows best.