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Ѕᴀᴀᴅ
  • Member for 8 years, 2 months
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30 votes

Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?

20 votes
Accepted

Proving $(λ^d + (1-λ^d)e^{(d-1)s})^{\frac{1}{1-d}}\leq\sum\limits_{n=0}^\infty\frac1{n!}λ^{\frac{(d^n-1)d}{d-1}+n}s^ne^{-λs}$

20 votes
Accepted

Calculate a quartic monic polynomial given that $P(1)=10,P(2)=20$ and $P(3)=30$.

16 votes
Accepted

Maximum of $\int_a^b \frac{f(x)}{x}\,\mathrm dx$

14 votes
Accepted

Prove $\lim\limits_{n\to∞}{\sum\limits_{x=0}^n\binom nx(1+{\rm e}^{-(x+1)})^{n+1}\over\sum\limits_{x=0}^n\binom nx(1+{\rm e}^{-x})^{n + 1}}=\frac 13$

12 votes
Accepted

Prove $\sum\limits_{n=0}^\infty\frac{5^n(3^{5^{n+1}}-5\cdot3^{5^n}+4)}{(729)^{5^n}-(243)^{5^n}-5\cdot3^{5^n}+1}=\frac12$

11 votes
Accepted

Evaluating $\lim_{n \to \infty}\left(\sqrt{n^2 - n+1}-\left\lfloor\sqrt {n^2 - n+1}\right\rfloor\right)$

10 votes

$x$-coordinate distribution on the $n$-sphere

10 votes
Accepted

If $a+b+c<0$ and $ax^2+bx+c=0$ has no real roots, is it true that $c$ must be less than $0$?

9 votes
Accepted

If $f(f(x))+f(x)=x^4+3x^2+3$, prove that $f$ is even

9 votes
Accepted

a simple proof that $\pi$ is irrational by Ivan Niven

8 votes
Accepted

What is the value of $f(100)$?

8 votes
Accepted

Prove that $\int_{2017}^{2018} (\arctan(\ln(x))-\ln(\arctan(x)))\,\mathrm dx>0$

8 votes
Accepted

Studying convergence of $\sum\limits_{k=1}^{\infty}\frac{k!\exp(k)}{(k+1)^k}$

8 votes
Accepted

Solving an Olympiad functional equation $f(f(n))=f(f(n+2)+2)=n$

8 votes
Accepted

Value bound for integrable periodic function

7 votes
Accepted

Minimal example of Simpson's paradox

7 votes

A simpler non-calculator proof for $17^{69}<10^{85}$

7 votes
Accepted

Does $\sum\limits_{k=1}^n\frac{a_i-a_k}{a_i+a_k}\cdot\frac{a_j-a_k}{a_j+a_k}=0$ for all $i\neq j$ imply $a_1=a_2=\cdots=a_n$?

7 votes
Accepted

Proving that the set of $a$ for which the roots of $P(X)-a$ are all real is an interval

7 votes
Accepted

Integral of error-like function

7 votes
Accepted

Maximum Entropy Distribution with Reciprocal Symmetry

7 votes
Accepted

Unbounded entropy solution to Burgers' equation

7 votes
Accepted

Finding the minimum of $λ$ such that $|a-b|^p\le λ(2-|a-b|)\left|\frac{a|a|^{p-1}}{1-|a|}-\frac{b|b|^{p-1}}{1-|b|}\right|$ for $a,b\in(-1,1)$

6 votes
Accepted

Integral identity using the transformation formula

6 votes

Does every integer $n > 2$ have a "reciprocating" pseudoprime $a$?

6 votes

Integer solution

6 votes
Accepted

Can every odd integer greater than $1$ be written as a product of fractions $\frac{4m+1}{2m+1}$?

5 votes
Accepted

Understanding $\mathcal F_\tau$ and showing that $X_\tau$ is $\mathcal F_\tau$-measurable where $\tau$ is a stopping time.

5 votes

General formula for $f(n)$

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