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In my obsevation: If $f(x)$ and $g(x)$ be two continuous funcions, and they have derivative and second derivative are also continuous in interval $[a, b]$. If $f(x)$ is more curvature than $g(x)$ in …

Let $ABCD$ be a quadrilateral, $P$ be a point in the plane let $E$, $F$ be the projections of the incenters of triangles $\triangle CPB$, $\triangle BPA$ onto $PB$ respectively; Let $G$, $H$ be the …

When I read the new paper 100 CHARACTERIZATIONS OF TANGENTIAL QUADRILATERALS-section 7, I remember that I posed some problem associated with tangential quadrilateral from 2014 in here and 2015 in here …

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 …

$\DeclareMathOperator\Area{Area}\DeclareMathOperator\cotg{cotg}$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 …

I am looking for a proof or a reference request for a problem as follows: Problem: Let a cyclic hexagon with sidelines $l_1$, $l_2$, $l_3$, $l_4$, $l_5$, $l_6$ and $l_1 \cap l_4 =A$, $l_3 \cap l_6 = …

I proposed a generalization of the Archimedean circle : In this figure $M$ is the midpoint of $AB$, $DE$; $(G)$, $(H)$, $(M)$ are the semicircles. Then two yellow circles are congruent. Question: Is t …

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 …

$\DeclareMathOperator\rad{rad}$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 …

Let $n > 1$ and $m > 0$ be two integers and $P_n$ be the $n^{th}$ prime. Prove: $$P_{n+m} \ge P_n + P_m .$$ Can you give a hint, reference, comment, or proof?

I am looking for a comment, reference, remark, or proof of three conjectures as follows: Conjecture 1: Let $x$ be an odd positive integer. Then there exist two integers $n, m \ge 2$ so that $$x=P_{ …

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 …

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 a hexagon $AB'CA'BC'$ let $AB' \cap A'B=C''$, $BC' \cap B'C = A''$, $CA' \cap C'A = B''$ then three conditions as follows equivalent: Three lines $AA', BB', CC'$ are concurrent (let the point o …

When talking about the condition for the three diagonals of a hexagon to be concurrent, we will think of Brianchon's theorem. Using software, I discovered a necessary and sufficient trigonometric rela …