Periodic functions (Romania 1996)

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Periodic functions (Romania 1996)

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Source: Romanian National Olympiad 1996 – Grade 11 – Problem 2

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#1 h Apr 13, 2025, 11:35 AM

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a) Let $f_1,f_2,\ldots,f_n: \mathbb{R} \to \mathbb{R}$ be periodic functions such that the function $f: \mathbb{R} \to \mathbb{R},$ $f=f_1+f_2+\ldots+f_n$ has finite limit at $\infty.$ Prove that $f$ is constant.

b) If $a_1,a_2,a_3$ are real numbers such that $a_1 \cos(a_1x) + a_2 \cos (a_2x) + a_3 \cos(a_3x) \ge 0$ for every $x \in \mathbb{R},$ then $a_1a_2a_3=0.$

This post has been edited 2 times. Last edited by Filipjack, Apr 13, 2025, 12:07 PM

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#2 h Apr 13, 2025, 12:30 PM

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a) With induction. Suppose $f$ is periodic with period $p>0$ and $f(\infty)=c$ . Then $f(x)=f(x+kp)\rightarrow c$ as $k\rightarrow \infty$ . Hence $f\equiv c$ . Now suppose $n>1$ , that it's true for $1,n-1$ and let $p>0$ be a period for $f_n$ .
Then $g(x):=f(x+p)-f(x)=\sum_1^{n-1} (f_k(x+p)-f_k(x))$ has limit 0 as $x\rightarrow \infty$ and each $f_k(x+p)-f_k(x)$ is periodic. Hence $g$ is constant, $g\equiv 0$ and $f$ has period $p$ . By the $n=1$ case, we find that $f$ is constant.
b) Set $f=\sum_1^3 \sin a_kx$ then $f'\ge 0$ everywhere and $f$ is bounded. Hence $f$ has a finite limit at $\infty$ and $f$ is constant. Since $f$ is odd, we have $\sum_1^3 \sin a_k x\equiv 0$ .
Hence we consider the case $\sin ax+\sin bx=\sin cx$ for all $x$ with $abc\not=0$ .
Taking derivatives in $x=0$ we find $a+b=c$ . Also $2\sin (cx/2)\cos ((a-b)x/2)=2\sin (cx/2)\cos (cx/2)$ for all $x$ . Hence $\cos ((a-b)x/2)=\cos(cx/2)$ for all $x$ by continuity. Taking double derivatives in $x=0$ we find $|a-b|=|c|$ . If $a-b=c$ then $a=b+c=c-b\implies b=0$ . If $b-a=c$ then $b=a+c=c-a\implies a=0$ . Contradiction.

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