Questions on proving, manipulating and applying inequalities.

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17
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1answer
193 views
+50

Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$

This question is strongly inspired by The smallest integer whose digit sum is larger than that of its cube? by Bernardo Recamán Santos. The number $n=124499$ has digit sum $1+2+4+4+9+9=29$ while its ...
1
vote
0answers
14 views
+300

Proof of multivariate regression plane maximizes correlation in normals

I am doing a homework sheet as practice for an upcoming course in multivariate statistics and been stuck on the following problem: Let ...
13
votes
1answer
285 views
+50

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...