Problems(2)
Elax creates a partially filled 4×4 grid, and is trying to write in positive integers such that any four cells that share no rows and columns always sum to a number S. Given that the sum of the numbers in the top row is also S, what is the missing cell number?[asy]
size(100);
add(grid(4,4));
label("11", (0.5,1.5));
label("10", (0.5,2.5));
label("?", (0.5,3.5));
label("8", (1.5,3.5));
label("7", (2.5,2.5));
label("4", (3.5,0.5));
label("9", (3.5,1.5));
[/asy]<spanclass=′latex−bold′>(A)</span>4<spanclass=′latex−bold′>(B)</span>6<spanclass=′latex−bold′>(C)</span>9<spanclass=′latex−bold′>(D)</span>10<spanclass=′latex−bold′>(E)</span>12 2022 A Individual
Friends Alice, Betty, and Cathy are playing a game. Betty and Cathy are each given a square number, such that Betty knows Cathy's number and Cathy knows Betty's, but neither of them know their own. Alice then says: "The sum of the numbers is less than 100."
Betty says: "If Cathy knew the number of possibilities for my number, she would know her own."
Cathy then says: "Now I know my number."What is Cathy's number?<spanclass=′latex−bold′>(A)</span>16<spanclass=′latex−bold′>(B)</span>25<spanclass=′latex−bold′>(C)</span>36<spanclass=′latex−bold′>(D)</span>49<spanclass=′latex−bold′>(E)</span>64
2022 B Individual