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Problem 888

AMC 12 late, AIME early

Combinatorics Difficulty 4.7 Prove it Brazilian Mathematical Olympiad · 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 $B$ ?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official 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 $x$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.