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Using my computer I discovered that: if $a,b,c$ are sidelengths of a triangle, then $(a+b-c)^\alpha+(b+c-a)^\alpha+(c+a-b)^\alpha \ge a^\alpha+b^\alpha+c^\alpha $ if $\alpha >1$ $(a+b-c)^\alpha+(b …

Rearrangement inequality: Assume we have finite ordered sequences of nonnegative real numbers $0 \le a_1 \le a_2 \le\cdots\le a_n \quad\text{and}\quad 0\le b_1 \le b_2 \le\cdots\le b_n, \cdots\,, \qua …

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 …

My question: Are the conjectures as follows correct? Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_k^{a_k}$$. …

$\DeclareMathOperator\rad{rad}$Conjecture: If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$. If the conj …

I improve my previous question. Because this conjecture is exactly natural development of A Muirhead Like Inequality and Muirhead's Inequality so I think the conjecture is true. But I can not prove it …

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 …

Four years ago, I proposed an inequality related to area and sides of a polygon. After computer checking, I conjecture that the previous inequality can be strengthened as follows: Let $A_1A_2\cdots A …

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: 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 …

Conjecture: Let $a_1, a_2, \cdots , a_n>0$ and $y \ge x $ then $$(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$$ Equality iff $x=y$ Is the conjecture right? Have you ever seen this in …

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 …

Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k). …

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: $ …

Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following: Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is …