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Symmetry. Put them all together and tell them to multiply forever. The question then becomes whether the $n$-th boy was born at the $2n$-th birth or later, or not, i.e., who is the majority among the …

Here is my computation. First of all, I had no idea whatsoever how to integrate implicit trigonometric functions, so I decided to switch to the integration with respect to the sides $x,y,z$ of the tri …

OK, having spent about 20 hours on the search of a nice proof (which extinguished my passion for beauty for the next several days at least), I'm resorting to the brute force. I will love to see someon …

I'm not in my best shape at the moment, so, please, check thoroughly what is written below. The answer is "No". The distribution does not matter much, but the norm does. So, we want to take $N\ge 3$ …

The argument below is not very elegant,but it is, indeed, a standard exercise. Let $g=\max(f-1,0)$. We shall prove that $$ f\log f\le 2g+\frac 2{p-1}g^p\,. $$ The integration and Holder then give the …

please show me how to prove for $|\omega|>20$ With great pleasure. We shall just show that $F(y)=\int_0^\infty e^{-x^a}x^a\log x\cos(yx)\,dx>0$ for large enough $y>0$. Note that $\cos(yx)=\Re e^{iyx}$ …

It is always true. Split $x_i=y_i+z_i$ where $y_i$ are bounded and $Ez_i\le \frac \mu{10n}$. You have no problems with $y_i$ because if they were alone,$ES_j$ would be concentrated in a very strong se …

Edit: I made it a bit clearer and simpler. No. You start with noticing that there is no "linear" estimate for the mean of $S_N$ in terms of the mean of the sample $X$ under the assumption that the m …

This should really be a comment, but it just takes too much space to put in the comment box. I was quite puzzled by Ryan's remark that Feige's problem (with just some constant) is hard while it is a t …

It is actually more like $e^{-\sqrt n}$. Let's look at the norm of the inverse matrix. The entries are $\pm\prod_{i:i\ne j}\frac 1{z_j-z_i}\sigma_m(z_1,\dots,z_{j-1},z_{j+1},\dots,z_n)$ where $z_k=e^{ …

OK, here is my argument (sorry for the delay). First of all, $Z$ is essentially the maximum of the absolute value of the polynomial $P(z)=\prod_j(1-z_jz)$ on the unit circumference (up to a factor of …

OK, let's try. If I haven't made any mistake, this is just a standard "condition on the head-cut the tail" measure theory puzzle. Claim 1. The condition $A_1$ that the sequence contains an arbitraril …

That is just Cuchy-Schwartz (though pretty well hidden). Writing everything down in terms of the joint density, dropping irrelevant factors, and taking into account that $f(-t)=-f(t)$, we see that the …

The answer to the first question is "certainly not". Consider the polynomials $P_d(X)=X^d-4X^{d-1}-X^{d-2}-1$. They have $d-1$ roots in the unit disk by Rouche, so their positive real roots are Pisot …

Ok, Loïc Teyssier, Gerry Myerson, Nik Weaver, Stefan Waldmann, coudy, and a few other people will scold me badly for answering a question about a relatively simple exercise in undergraduate probabili …