There are two well known songs called All of Me. Belle Baker was the first to sing All of Me in 1931, and Billie Holiday did a great cover of it. It's a classic. There are two songs called Stay With Me. Lorraine Ellison was the first to sing Stay with Me in 1966. That is the version Sam Brown, Bette Midler, and others covered. Rod Stewart and Ronnie Wood wrote another one which Faces recorded in 1971. I remember both of those on the radio when I was young. Since you didn't specify a year, Baker, Ellison and Faces should also be accepted for those songs. If you are going to accept later covers, there were lots of singers who covered both of them.
all of these songs are from the billboard 100, and they are all recent, so I am pretty sure the quiz maker is not referring to songs of the last century.