He pays to sign up for a website then he answers questions to win a prize

This competition is a kind of gambling, and gambling is definitively forbidden by Allaah as He says (interpretation of the meaning): 

“O you who believe! Intoxicants (all kinds of alcoholic drinks), and gambling, and Al‑Ansaab (stone altars for sacrifices to idols etc), and Al‑Azlaam (arrows for seeking luck or decision) are an abomination of Shaytaan’s (Satan’s) handiwork. So avoid (strictly all) that (abomination) in order that you may be successful.

91. Shaytaan (Satan) wants only to excite enmity and hatred between you with intoxicants (alcoholic drinks) and gambling, and hinder you from the remembrance of Allaah and from As‑Salaah (the prayer). So, will you not then abstain?”

[al-Maa’idah 5:90-91]

Gambling – as al-Maawardi said – is that in which the participant will be either a winner if he takes, and a loser if he gives. End quote from al-Haadi al-Kabeer (15/192). 

This is the case in this transaction, because the participant will either lose the price of his subscription, or he will gain more if he answers the questions. 

Another reason why it is haraam is that the Prophet (peace and blessings of Allaah be upon him) said: “There should be no prizes except for archery, camel racing or horse racing.” Narrated by al-Tirmidhi (1700) and al-Nasaa’i (3585) 

The Prophet (peace and blessings of Allaah be upon him) mentioned these three because they are skills used in jihad for the sake of Allaah. 

Some scholars added to these three everything that helps with jihad and spreading Islam, such as competitions in reciting Qur’aan or in knowledge of hadeeth and fiqh, so it is permissible to award prizes for these. 

It is not permissible to enter competitions in which entrants pay money (the cost of subscribing or participating) then if one wins he takes the prize, except those which were mentioned by the Prophet (peace and blessings of Allaah be upon him) only. 

And Allaah knows best.