Competition problem — Combinatorics

Combinatorics Difficulty 4.7 AIME Prove it Brazil

Box $A$ contains black balls and box $B$ contains white balls. Take a certain number of balls from $A$ and place them in $B$ . Then take the same number of balls from $B$ and place them in $A$ . Is the number of white balls in $A$ then greater, equal to, or less than the number of black balls in $B$ ?

Solution

Suppose we move $n$ balls from $A$ to $B$ , then $x$ white balls from $B$ and $n-x$ black balls from $B$ . That leaves $x$ black balls in $B$ . So both numbers equal $x$ .

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