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Brazilian Mathematical Olympiad

Algebra Difficulty 4.5 AIME Prove it Brazil

Two players play a game as follows. The first player chooses two non-zero integers $A$ and $B$ . The second player forms a quadratic with $A$ , $B$ and $1998$ as coefficients (in any order). The first player wins iff the equation has two distinct rational roots. Show that the first player can always win.

Solution

Choose $A$ and $B$ such that $A + B + 1998 = 0$ . Then $1$ is a root of the quadratic equation no matter how the second player arranges the coefficients.

The other root is also rational, because the product of the roots, the quotient of two of the rational coefficients $A$ , $B$ , $1998$ , is rational.

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