Is it permissible to memorize the Quran with the intention of obtaining a prize, or with the intention of memorizing the Surah and the prize together?

It is not permissible for a Muslim to seek knowledge or memorize the Holy Quran with the intention of participating in competitions or gaining a worldly عرض (benefit/display), because Allah Almighty accepts only that work which is pure and sought for His countenance. So, whoever associates another with Allah in his work, Allah will abandon him and his shirk (association). The work must be purely for Allah Almighty. After that, there is no objection to participating in competitions. The Prophet, peace and blessings be upon him, warned that whoever learns knowledge by which the countenance of Allah Almighty is sought, but learns it only to attain a worldly عرض (benefit/display), will not find the fragrance of Paradise on the Day of Resurrection.