Problem 14 · 2019 Math Kangaroo
Hard
Algebra & Patterns
custom-operationsubstitution
Michael invents a new operation \(\diamond\) for real numbers, defined by \(x \diamond y = y - x\). Which of the following statements is definitely true if numbers \(a\), \(b\) and \(c\) satisfy \((a \diamond b) \diamond c = a \diamond (b \diamond c)\)?
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Answer: D — \(a = 0\)
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Hint 1 of 2
Carefully apply \(x \diamond y = y - x\) to each side of the equation.
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Hint 2 of 2
Both sides become a simple expression in \(a\), \(b\), \(c\); the difference shows which variable is forced.
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Approach: expand both sides of the condition
- With \(x \diamond y = y - x\), the left side is \((a \diamond b) \diamond c = c - (b - a) = c - b + a\).
- The right side is \(a \diamond (b \diamond c) = (c - b) - a = c - b - a\).
- Setting them equal gives \(c - b + a = c - b - a\), so \(2a = 0\), i.e. \(a = 0\).
- Answer (D) \(a = 0\).
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