Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear brother in Islam, we would like to thank you for the great confidence you place in us, and we implore Allah Almighty to help us serve His cause and render our work for His Sake.
With regard to your question, Dr. `Abdel-Fattaah Idrees, professor of Comparative Jurisprudence at Al-Azhar University, states:
Institutions, academic or whatever, award prizes to whoever provides the best answer or solution to a given problem or issue. Such awards must not be considered as a price for the benefit accrued from the prize-giving party. Rather, the aim is to reward those who worked hard with efforts that speak on their behalf. Prizes are awarded to motivate and encourage others to produce work that is for the progress of society.
The Prophet (peace and blessings be upon him) was the foremost in staging competition between his Companions. He did so by asking to answer questions. `Abdullah ibn `Umar (may Allah be pleased with him) narrates that the Prophet (peace and blessings be upon him) said: “Among trees there is one whose leaves never fall; it typifies a true Muslim (in sticking to the noble values of Islam). Can any one of you tell me what it is?” This question put the people off balance, and they all said it was a tree in the desert. I (`Abdullah ibn `Umar) thought it was a palm tree, but I felt too shy to say so (in the presence of notable Companions like Abu Bakr and `Umar). Failing to answer it, the people said, “Tell us what is it, O Prophet.” He said, “It is palm tree.”
According to the majority of scholars, what makes it permissible to compete to find the correct or best answer for any given question, academic or non-academic, is the noble cause on which it is based.
So, by and large, it is quite clear that such competition is permissible in an Islamic perspective on the condition that no prize is awarded. This is according to the views of the majority of scholars.
However, the Hanafi School maintains that awarding prizes is permissible, because they will act as a stimulus and incentive for others to work harder.
Based on this latter view, it is permissible in Islam for state institutions to award prizes for the best performance, as this will encourage people to put more effort into developing their career and will ultimately lead to the progress of the whole community.
Almighty Allah knows best.