Explain why Madame and Pierre Curie were able to discover radium, even though its half-life of 1603 years is much less than what is usually considered to be the age of the universe.

Wikipedia (including footnotes and a bibliography) says,

Radium (Latin radius, ray) was discovered by Marie Skłodowska-Curie and her husband Pierre in 1898 in pitchblende coming from North Bohemia, in the Czech Republic (area around Jáchymov). While studying pitchblende the Curies removed uranium from it and found that the remaining material was still radioactive. They then separated out a radioactive mixture consisting mostly of barium which gave a brilliant green flame color and crimson carmine spectral lines which had never been documented before. The Curies announced their discovery to the French Academy of Sciences on 26 December 1898.[5]

As for the half-life part of your (homework) question, you will have to Google for that yourself.