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Redundant Aunt
  • Member for 8 years, 9 months
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27 votes

Proving that a function is odd

17 votes

The inequality. Regional olympiad 2015

13 votes

Evaluate the series $\sum_{n=1}^{\infty} \frac{2^{[\sqrt{n}]}+2^{-[\sqrt{n}]}}{2^n}$

12 votes
Accepted

Prove the inequality $a^3+2 \geq a^2+2 \sqrt{a}$

11 votes
Accepted

Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root.

11 votes
Accepted

How to solve the functional equation $f\left(x^2+f(y)\right)=y+f(x)^2$

9 votes
Accepted

Limit involving binomial coefficients: $\lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}}$

8 votes
Accepted

$f(xf(y)-f(x))=2f(x)+xy$

8 votes
Accepted

Proving $f: \Bbb{Z} \times\Bbb{Z} \to \Bbb{Z}, \ f(a,b)=3a-2b$ is surjective without using linear diophantine equation

8 votes

How to solve $f(x+f(x)+2f(y))=f(2x)+f(2y)$?

7 votes

How to calculate $ \lim_{n\to \infty}\left(\sum_{k=1}^{n}\frac{1}{3k+1}-\frac{\ln n}{3}\right)$

7 votes
Accepted

How to compute fraction sums?

7 votes
Accepted

Limit of $n^{\phi(n)}$ where $\phi(n) \rightarrow 0$?

7 votes

Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

7 votes

Suppose $f$ is differentiable on $\mathbb{R}$ and that $\lim_{x \rightarrow 0} f'(x)=L$. May we conclude that $f'(0)=L$

7 votes

Fibonacci sequence starting with any pair of numbers

7 votes

Limit $\lim_{n\to\infty}\sum_{k=0}^n\binom nk\frac{3k}{2^n(n+3k)}$

7 votes
Accepted

Solving functional equation $f(x+y)^2=f(x)^2+f(y)^2$

7 votes

Inequality $(x-1)(y-1)(z-1)\geq 8$ provided that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$

6 votes
Accepted

Prove that for all real numbers $x,y$ and $z$ that $x^2+y^2+1 \geq x+xy$.

6 votes
Accepted

A problem involving the product $\prod_{k=1}^{n} k^{\mu(k)}$, where $\mu$ denotes the Möbius function

6 votes

Convergence of $\sum_{n=1}^{\infty}{\frac{(n-1)^n}{n^{n+1}}}$

6 votes

If $S=\{1,2,\dots,n\}$ and $P_n(k)$ be the number of permutations of $S$ having $k$ fixed points, then $\sum_{k=0}^nk.P_{n}(k)=n!$

6 votes
Accepted

Can the pre-image (under homomorphism) of a subgroup be empty?

6 votes

Evaluate $\int_0^{\pi/2}\log\cos(x)\,\mathrm{d}x$

5 votes

Let $a,b,c,d>0$ and $a+b+c+d=1$. Prove that $\frac{abc}{1+bc}+\frac{bcd}{1+cd}+\frac{cda}{1+ad}+\frac{dab}{1+ab}\le \frac{1}{17}$

5 votes

Let $x,y,z>0,xyz=1$. Prove that $\frac{x^3}{(1+y)(1+z)}+\frac{y^3}{(1+x)(1+z)}+\frac{z^3}{(1+x)(1+y)}\ge \frac34$

5 votes

Olympiad inequality (Cauchy/AM-GM sort)

5 votes

$\prod_{i=1}^{n-1} a_i = 1 \Rightarrow \prod_{i=1}^{n-1} (1+ a_i)^{i+1} > n^n$?

5 votes

Solve in non-negative integers: $m^2+n^2=1997 (m-n)$

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