Baltic Way 1990 · Problem 5

Algebra

Let $*$ denote an operation, assigning a real number $a * b$ to each pair of real numbers ( $a, b)$ (e.g., $a * b=$ $a+b^{2}-17$ ). Devise an equation which is true (for all possible values of variables) provided the operation $*$ is commutative or associative and which can be false otherwise.

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Topics

Algebraic manipulation

Solutions

Solution

Solution:

A suitable equation is $x * (x * x) = (x * x) * x$ which is obviously true if $*$ is any commutative or associative operation but does not hold in general, e.g., $1 - (1 - 1) \neq (1 - 1) - 1$ .

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AoPS · Baltic Way 1990