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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 …

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 $A'$, $B'$ …

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 …

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 …

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 …

Question: Are the properties as follows holds? Version 1: the answer by Bjørn Kjos-Hanssen Let $P$ be a positive integers. We written: $P=$ $a_1^{x_1}a_2^{x_2}...a_n^{x_n}$ $=b_1^{y_1}b_ …

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 …

Ono's inequality is true for acute triangle but false with general triangles. The inequality as follows is false with general triangls but I think it true with acute triangle (follows answer by Fedor …

Is the inequality as follows true? Let $k > 0$, $a_i$ Is a complex number for $1\le i\le n$ and let $$S:=a_1+a_2+....+a_n$$ Suppose that $$b_i:=S-ka_i \quad\text{ for} \quad 1\le i\le n.$ …

Padoa's inequality is named after Alessandro Padoa (1868-1937): Let $a$, $b$, $c$ be sidelengths of a given triangle $\triangle ABC$ then $$(b+c-a)(c+a-b)(a+b-c) \le abc .$$ My question: Is …

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 …

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 …

From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference. Inequality: Let $y=f …

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 …

By my computation, I pose a conjecture as follows and I am looking for a proof: Conjecture: Let $(O)$ be a circle with radius $R$, and $n$ be positive integer $n\ge 3$. Construct $n$ circles …