Competition problem — Combinatorics

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Combinatorics Difficulty 5.6 AIME, harder Prove it Soviet Union

Problem:

Two players have a $3 \times 3$ board. $9$ cards, each with a different number, are placed face up in front of the players. Each player in turn takes a card and places it on the board until all the cards have been played. The first player wins if the sum of the numbers in the first and third rows is greater than the sum in the first and third columns, loses if it is less, and draws if the sums are equal. Which player wins and what is the winning strategy?

Solution

Solution:

The first player always wins.

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