In the magazine Gazeta Matematică 11/1978, page 494.
I called problem 17503 "problem brew" as a play on words in harmony with the Breaza , locality where the author is from.
"17503*. The polynomial $P$ has real and distinct roots. If $x_0$ is a root of it,
prove the inequality
$3(P''(x_0))^2-4P'(x_0)\cdot P'''(x_0)\geqslant 0$.
{Author : } N. TOMA, teacher, Breaza"
Solution CiP
Let $x_0,\;x_1,\;x_2,\;\dots \;,x_m\;\in \mathbb{R}$ be the roots of the polynomial $P$. Then
$P(x)=(x-x_0)\cdot \color{Blue}Q(x)=(x-x_0)\cdot \color{Blue}{\alpha (x-x_1)(x-x_2)\dots (x-x_m)} \tag{1}$
We have $P(x_0)=0$ and $P'(x)=Q(x)+(x-x_0)Q'(x)$ , so
$P'(x_0)=Q(x_0) \tag{2}$
Further, $P''(x)=Q'(x)+Q'(x)+(x-x_0)Q''(x)$ so
$P''(x_0)=2Q'(x_0) \tag{3}$
Deriving once again $P'''(x)=2Q''(x)+Q''(x)+(x-x_0)Q'''(x)$ hence
$P'''(x_0)=3Q''(x_0) \tag{4}$
The quantity to be proven to be nonnegative is $J=3(P''(x_0))^2-4P'(x_0)\cdot P'''(x_0)\;\underset{(3)\;(4)}{\overset{(2)}{=}}\;3\cdot 4(Q'(x_0))^2-4Q(x_0)\cdot 3Q''(x_0)$ so
$J=12[(Q'(x_0))^2-Q(x_0)Q''(x_0)] \tag{5}$
But, with $Q(x)$ given by (1) we have
$$Q'(x)=\sum_{i=1}^m \left (\alpha \cdot \prod_{k\neq i}^{k\in \{1,\;2,\;\dots m\}}(x-x_k)\right )=\sum_{i=1}^m\left (\alpha \cdot \frac{\prod_{k=1}^m(x-x_k)}{x-x_i}\right )=\sum_{i=1}^m\frac{Q(x)}{x-x_i}$$
whence
$$\frac{Q'(x)}{Q(x)}=\sum_{i=1}^m\frac{1}{x-x_i}$$
and deriving this equality we obtain
$$\frac{Q''(x)\cdot Q(x)-(Q'(x))^2}{(Q(x))^2}=-\sum_{i=1}^m\frac{1}{(x-x_i)^2}$$
Setting $x=x_0$ above we find
$$(Q'(x_0))^2-Q(x_0)\cdot Q''(x_0)=(Q(x_0))^2\cdot \sum_{i=1}^m\frac{1}{(x_0-x_i)^2}\geqslant 0$$
which shows us, using (5) , that $J\geqslant 0$
$\blacksquare$
The second problem is 17498 whose author has the name of a hen. Hence the expression "In the mind of a rooster"...
"17498*. For any $x\in \mathbb{R}$ , let $u_n(x)=\underset{n\;times}{\underbrace{\sin \sin \dots \sin x}}$ .
Prove that the series $$\sum_{n=1}^{\infty}u_n(x)$$ has a sum for any $x\in\mathbb{R}$ and determine its sum.
{ Author : } Stelian GĂINĂ, Bucharest"
WRONG ANSWER CiP
$$\sum_{n=1}^{\infty}u_n(x)=$$???
ANSWER CiP
$$\sum_{n=1}^{\infty}u_n(x)=\begin{cases}+\infty\;\;\;if\;x>0\\0\;\;\;\;if\;x=0\\-\infty\;\;if\;x<0 \end{cases}$$
Solution CiP
Let us note, for convenience, $\sin_n(x)=\underset{n\;times}{\underbrace{\sin \sin\dots \sin (x)}}$. Then our series is written :
$$s(x):=\sum_{n=1}^{\infty}\sin_n(x)=\sin x+\sum_{n=2}^{\infty}\sin_n(x)=\sin x+s(\sin x)\tag{10}$$
where do we get the relationship :
$s(x)-s(\sin x)=\sin x \tag{11}$
We put in (11) $\sin x$ instead of $x$ :
$s(\sin x)-s(\sin_2 x)=\sin_2 x \tag{12}$
same as above
$s(\sin_2 x)-s(\sin_3 x)=\sin_3 x \tag{13}$
and so on
............................................................................
$s(\sin_{n-1} x)-s(\sin_n x)=\sin_n x \tag{1n}$
This is where the author of the solution got stuck, not knowing what to do...I wanted to summing these relationships, but the path seems to lead to no success.
The path followed cannot succeed because the transition from (10) to (11) is illegal , the series being divergent as we will show below.
Although $\sin_n x \xrightarrow [n\to\infty]{}0$ , we have $\sqrt{n}\sin_n x \xrightarrow [x>0]{n\to \infty} \sqrt{3}$ so
$\sin_n x \sim \frac{\sqrt{3}}{\sqrt{n}}$
and since the series $\sum_{n\geqslant 1}\frac{1}{\sqrt{n}}$ diverges at $+\infty$ , so will the series $\sum_{n\geqslant 1}u_n$.
The most general result known is :
$\displaystyle \lim_{n\to\infty}\sqrt{n}\sin_n x=\sqrt{3}\cdot sgn(\sin x) \tag{20}$
One source is the Polish magazine MATEMATYKA, no. 3 from 1987, pages 138-140, where a generalization is also presented. Or, the book in Romanian,
"TEODORESCU Nicolae (coord.)
Probleme din Gazeta Matematică: Ediție selectivă și metodologică
Ed Tehnică, București, 1984"
Problem SC6, pages 487-486 (although in a particular case).
I invite you to study one of the indicated sources.
$\blacksquare\;\blacksquare$
REMARK CiP It seems that the author of this blog has been walking around like a drunken chicken lately, making absurd posts. Initially he thought he could somehow solve the functional equation (11).
