Romanian National Olympiad 1997 - Grade 11 - Problem 3

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Romanian National Olympiad 1997 - Grade 11 - Problem 3

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Source: Romanian National Olympiad 1997 - Grade 11 - Problem 3

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#1 h Apr 6, 2025, 10:11 AM

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Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$

a) Prove that there exist nonconstant functions in $\mathcal{F}.$

b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.

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#2 h Apr 6, 2025, 12:44 PM

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Filipjack wrote:

Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$

a) Prove that there exist nonconstant functions in $\mathcal{F}.$

b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.

a) $f=\cos(\sin \frac x2)\xRightarrow{\hspace{40pt}}f\in\mathcal F.$
b) Assume towards a contradiction that $\exists f\in\mathcal F,\{x:f'=0\}$ is finite, choose its max element, say $a$ . Assume $\exists x_1, x_2>a: f'(x_1)<0, f'(x_2)>0$ . By Darboux Theorem, $\exists x_3: f'(x_3)=0$ , contradicting maximality of $a$ . This implies $f'$ either is always positive or negative $\forall x>a\xRightarrow{\hspace{40pt}} f$ is strictly increasing or decreasing $\forall x>a \xRightarrow{\hspace{40pt}}f(x+\sin x)>f(x)$ where $x>a, \sin x>0$ , when $f$ is strictly increasing and $x>a, \sin x<0$ when $f$ is strictly decreasing leading to a contradiction to $f(x)\ge f(x+\sin x).$

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#3 h Apr 7, 2025, 6:07 AM

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Filipjack wrote:

Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$

a) Prove that there exist nonconstant functions in $\mathcal{F}.$

b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.

You can also refer https://math.stackexchange.com/questions/1429915/fx-ge-fx-sin-x-nonconstant-functions-infinite-number-of-solutions-to?noredirect=1

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