What is the ruling on taking part in an online contest to win a prize of 100,000 dollars in return for answering thirty questions in different fields of knowledge and science? Knowing that there is no drawing of lots; only one or two cards are purchased to commence the contest.

Should a woman who broke her fast because of delivery make up for missed fasting days before the next Ramadhaan, and what is the expiation due on her in case she delayed making up for them ?

She should make up for missed fasting days before the start of next Ramadan if possible, but if she didn`t while being able to, then she is obliged to make up for them along with feeding a needy person for each delayed day of the missed fasting days. However, if she wasn`t able to make up for the missed fasting days before the start of next Ramadan, she has to fast a day for every day that she missed, and no ransom is due on her. And Allah Knows Best.

Is it permissible to repeat the Istikhara (seeking guidance) prayer, and is it a condition that one should sleep after offering it?

All perfect praise be to Allah,The Lord of The Worlds                                                                                                                                                            Istikhara is offered once, but it is permissible to repeat it, and sleeping after offering it is not a condition for its validity. And Allah Knows Best.

After I had prayed, I found a drop of menstrual blood and was unsure when that happened; what is the ruling on my prayer and fast?

All praise is due to Allah, peace and blessing be upon the Messenger of Allah.

Your prayer and fast are correct ,and you should not make up for them, and we ask Allah, The Almighty, to duely reward you for these acts of worship. Your menses start once you see blood because the situation remains as such unless there is certainty or predominant conjecture, as per the Fiqh maxim: "Certainty is not removed by doubt". And Allah Knows Best.