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Is the conjecture on A+B=C following correct ? Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write: $ …

Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to: \begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\s …

I am looking for a proof of the inequality as follows: Conjecture: Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$. Let $B_1B_2....B_n$ be a polygon incribed the ci …

I am looking for a proof of the inequality as follow: Let $n$ be an integer number $n \ge 2$ and $x_1, \cdots, x_n$ and $y_1,\cdots, y_n$ are nonegative real numbers such that $(x_1,\cdots, x_n)$ …

Generaliation the result in our paper for sum and similarly my previous question for product. I have a question: My question: Distributing $N$ points on the sphere so that the sum of their mutual dis …

$\DeclareMathOperator\Area{Area}\DeclareMathOperator\cotg{cotg}$I am looking for a proof (or a reference) of an inequality related to area and the sidelengths of a polygon as follows: Let $A_1A_2\cdo …

Let $P(n)$ be the statement that $$n < \mathrm{rad}(n(n-1)(n-2)),$$ where $\mathrm{rad}$ is the radical of an integer, that is defined as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ pri …

The Fejer-Jackson inequality as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$ I conjecture that the inequality as follows holds: $$\sum_{k=1}^n\f …

Let $C$ is a perimeter of a convex hull (plane geometry) and $d_{max}$ is the largest distance of two arbitrary points in the convex hull. I am looking for a proof that: $$\frac{C}{d_{max}} \le \pi …

I have posed conjectures of two inequalities as follows: Inequality 1: Let $n>2$ and $1 \le m \le n$ be integers. … {a_3}^{y_3}$$ My question: I am looking for a proof of two inequalities above. …

Update: A year ago, but the first answer is not clear with me. I bounty this question again. My question: I am looking for a proof or counterexample to the following inequality: If $n \in \mathb …

Since Brahmagupta's formula and Bretschneider's formula we have the inequality: Any two quardrilaterals $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ with the same sidelengths and $A_1A_2A_3A_4$ is a cyclic b …

Let $ABC$ be arbitrary triangle, $D$, $E$, $F$ are the midpoints of $BC$, $CA$, $AB$ respectively. Define points, segments in the figure below. I am looking for a proof that: $$DE+EF+FD \le (DG+DH+E …

More than seven years ago. I posted this problem in stackexchange: Let $ABC$ be a triangle, $P$ be arbitrary point inside of $ABC$. Let $A_1B_1C_1$ be the tangential traingle of $ABC$. Let $A'$, $B'$ …

I am looking for a proof of the inequality as follows: Let $A_1A_2....A_n$ be the regular polygon incribed in a circle $(O)$ with radius $R$. Let $B_1B_2....B_n$ be a polygon incribed the cir …