# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### How does one prove the matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
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### Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing

Prove without calculus that the sequence $$L_{n}=\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}, \space n\in \mathbb N$$ is strictly decreasing.
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### Prove $\left(\dfrac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

Inadvertently, I find this interesting inequality,But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find ...
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### How can one prove that $e<\pi$?

This question is inspired by another one, asking to prove that something approximately equal to $1.2$ is bigger than something approximately equal to $0.9$. The numerical answer to this question was (...
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### What is Cauchy Schwarz in 8th grade terms?

I'm an 8th grader. After browsing aops.com, a math contest website, I've seen a lot of problems solved by Cauchy Schwarz. I'm only in geometry (have not started learning trigonometry yet). So can ...
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### Fastest way to check if $x^y > y^x$?

What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that? The issue is that $x$ and $y$ can be very large.
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### A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
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### A proof of the Isoperimetric Inequality - how does it work?

Here is a nice proof of the isoperimetric inequality (equality part ommited): Isoperimetric Inequality If $\gamma$ is any simple closed piecewise $C^1$ curve of length $l$, with it's interior having ...
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### Which one is bigger: $\;35{,}043 × 25{,}430\,$ or $\,35{,}430 × 25{,}043\;$?

Which of the two quantities is greater? Quantity A: $\;\;35{,}043 × 25{,}430$ Quantity B: $\;\;35{,}430 × 25{,}043$ What is the best and quickest way to get the answer without using ...
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### Why does this way of solving inequalities work?

Here is what I had to prove. Question: For positive reals $a$ and $b$ prove that $a^2+b^2 \geq 2ab$. Here is how my teacher did it: First assume that it is in fact, true that $a^2+b^2 \geq 2ab$....
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### A combinatorial proof of $n^n(n+2)^{n+1}>(n+1)^{2n+1}$?

The statement is simply that the sequence $\left(1+\frac{1}{n}\right)^n$ is increasing. Since the numbers $n^m$ have quite natural combinatorial interpretations, it makes me wonder if a ...
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### Which is bigger: $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$ or $9!!!!!!!!!$?

In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) ...
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### Olympiad Inequality $\sum_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the ...
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### If both $a,b>0$, then $a^ab^b \ge a^bb^a$

Prove that $a^a \ b^b \ge a^b \ b^a$, if both $a$ and $b$ are positive.
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### $x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $x^2y+y^2z+z^2x < \frac12$

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $$x^2y+y^2z+z^2x < \frac12$$ This inequality has been verified by Mathematica. $\frac12$ is not the best bound. I try to do AM-GM for this one but not ...
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### Prove that $a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}$

Let $a$, $b$ and $c$ be non-negative numbers. Prove that: $$a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}.$$ I have a proof, but my proof is very ugly: it's ...
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### Proving the inequality $\frac{\log (1)}{1!}+\frac{\log ^2(2)}{2!}+\frac{\log^3(3)}{3!}+\cdots> \frac{\pi }{4}$

How to prove this inequality? $$\frac{\log (1)}{1!}+\frac{\log ^2(2)}{2!}+\frac{\log^3(3)}{3!}+\cdots> \frac{\pi }{4}$$ The left side looks vaguely like the series for $\exp(x)$: the terms ...
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In trying to understand a result of D. Rider (Trans. AMS, 1973) I've got stuck on a lemma that he uses. At one point he makes a step without comment or explanation, but I can't see why it works. Here ...
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### What is the larger of the two numbers?

What is the larger of the two numbers? $$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$ I solved this, and I think that is an interesting elementary problem. I want different points ...
### How prove this geometry inequality $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2$
Zhautykov Olympiad 2015 problem 6 This links discusses the olympiad problem which none of students could solve , meaning it is very hard. Question: The area of a convex pentagon $ABCDE$ is $S$, ...