Souvik Dey
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### Questions (267)

 21 Evaluate $\int_{0}^1 \prod_{k=2}^n \lfloor kx \rfloor dx$ 14 If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x)\neq x$ for all $x$, must it be true that $f(f(x))\neq x$ for all $x$? 12 Verifying the inequality $\sum_{i=1}^n \frac{\cos y_i}{\sin x_i}≤ \sum_{i=1}^n\cot x_i$ 11 $x,y,z$ positive real numbers , $x+y+z=3$ $\implies x^4y^4z^4(x^3+y^3+z^3)≤3$ 10 Partition of $\mathbb{R}^+$ into two semigroups

### Reputation (3,497)

 +5 $H:=\{ g^2 : g \in G \}$ is a subgroup of $G$ $\implies$ $H$ is normal in $G$ -2 Can a non-discrete metric $d$ be always defined on any non-empty set $X$ such that $(X,d)$ becomes a complete metric space? +5 If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x)\neq x$ for all $x$, must it be true that $f(f(x))\neq x$ for all $x$? -2 $a,b,c>0$ , $a^a b^b c^c =1$ then $a+b+c \le 3$?

 8 $f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) )$ 6 Primes and Twine primes and their sums. 5 Need hint: Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent. 4 Simple inequality on real numbers 3 Integer Solutions to $x^2+y^2=5z^2$

### Tags (117)

 10 functional-equations × 6 7 algebra-precalculus × 11 8 prime-numbers × 12 6 prime-twins 7 number-theory × 37 4 elementary-number-theory × 8 7 sequences-and-series × 26 3 abstract-algebra × 17 7 inequality × 21 3 diophantine-equations × 7

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 Mathematics 3,497 rep 1841 Physics 106 rep 2