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The answer of Iosif helps one try to change variables as the solution verify $x^2+y^2=1$. This one is a bit long but oddly it has many powerful factorisation arguments. So set $\sqrt{x^2+y^2}=a$ and $ …

I came across the following while doing some related proof; It seems easy to prove. $\quad$ We are in ${\mathbb{M}}_n(\mathbb{C})$, $n>1$: $1$) Given a unitary $n\times n$ matrix $U$, there is some …

Here is a way, as noted, if $F'(1)<0$ we find one root in $(1,+\infty)$ (proved later). Assume $F'(1)\ge 0$, so necessarily $0<a<\dfrac{1}{3}$, for $[(2-a)(3a -1)p^2+4(a+2)p+10]\le 0$, $p$ should be o …

I am just putting an (incomplete) answer as i think this can be done in the following ''elementary'' way. And also since this would be giving some nice induction proofs... For a visualisation one may …

Here is a classic way of proving this inequality for a triangle. We fix our circle $(C)$ and take the triangle $ABC$ inscribed in $(C)$. Fix a side say $BC$ we want to prove that moving the point $A …

I post this instead of a comment, feel free to comment. The inequality in the main question for $p=1$ (and $ p>1$) can be found in chap. 3 Topics in matrix analysis by R. Horn and C. Johnson, (i don …

I guess this type of inequality can be fast proven as follows: Notice that the expression $\det(A-B)$ is an affine function in the entries of $A$ and $B$ considered as variables. Each row of $A=[a_{i, …