Problem 28 · 2024 Math Kangaroo
Stretch
Algebra & Patterns
symmetrysum-constraint
A function \(f:\mathbb{R}\to\mathbb{R}\) fulfils the condition \(f(20-x)=f(22+x)\) for all real numbers x. It is known that f has exactly two real zeros. What is the sum of the two zeros?
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Answer: E — another number
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Hint 1 of 2
The condition says f takes equal values at points that are mirror images of each other.
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Hint 2 of 2
Find the axis of symmetry by averaging 20 − x and 22 + x; the two zeros are symmetric about it.
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Approach: locate the axis of symmetry
- f(20 − x) = f(22 + x) means f is symmetric about the midpoint of 20−x and 22+x.
- That midpoint is (20−x + 22+x)/2 = 21, so the graph is symmetric about x = 21.
- Two zeros symmetric about 21 add to 2×21 = 42, which is another number.
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