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Continuity of function and line segment of integer length
egxa   4
N May 8, 2025 by jasperE3
Source: All Russian 2025 11.8
Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of \( f \). It is known that the graph of \( f \) contains exactly \( N \) chords, one of which has length 2025. Find the minimum possible value of \( N \).
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egxa
Apr 18, 2025
jasperE3
May 8, 2025
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Continuity of function and line segment of integer length
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Source: All Russian 2025 11.8
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egxa
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egxa
#1 h  Apr 18, 2025, 5:17 PM
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Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of \( f \). It is known that the graph of \( f \) contains exactly \( N \) chords, one of which has length 2025. Find the minimum possible value of \( N \).
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tonykuncheng
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tonykuncheng
#2 h  Apr 18, 2025, 5:38 PM
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is it $1$?
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NO_SQUARES
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NO_SQUARES
#3 h  Apr 23, 2025, 8:10 AM
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tonykuncheng wrote:
is it $1$?

No, answer is such.
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arzhang2001
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arzhang2001
#4 h  May 8, 2025, 2:54 PM
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wired problem! clearly there is example with just 1 chord with length 2025. consider this:
$|x|+|x-1|$ this is obviously a continues function with just one chord with length 2025
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jasperE3
11354 posts
jasperE3
#5 h  May 8, 2025, 4:38 PM
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arzhang2001 wrote:
wired problem! clearly there is example with just 1 chord with length 2025. consider this:
$|x|+|x-1|$ this is obviously a continues function with just one chord with length 2025

There is more than one chord here, for instance a chord of length $1$ between $(0,1)$ and $(1,1)$.
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