Michael Rozenberg's user avatar
Michael Rozenberg's user avatar
Michael Rozenberg's user avatar
Michael Rozenberg
  • Member for 7 years, 10 months
  • Last seen this week
  • Tel-Aviv
98 votes
12 answers
60k views

References for multivariable calculus

94 votes
2 answers
6k views

Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing

  • 43.3k
86 votes
6 answers
40k views

Cardinality of set of real continuous functions

  • 7,365
85 votes
7 answers
10k views

If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$

76 votes
10 answers
44k views

Examples of finite nonabelian groups.

63 votes
6 answers
20k views

Strategies to denest nested radicals $\sqrt{a+b\sqrt{c}}$

  • 13k
54 votes
8 answers
119k views

Odd/Even Permutations

  • 1,085
47 votes
5 answers
2k views

For $a$, $b$, $c$, $d$ the sides of a quadrilateral, show $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. (A generalization of IMO 1983 problem 6)

44 votes
2 answers
3k views

Prove that $a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}$

40 votes
5 answers
2k views

Prove that $(1+x)^\frac{1}{x}+(1+\frac{1}{x})^x \leq 4$

  • 15.5k
40 votes
8 answers
2k views

Show $\frac{\sin x_1\sin x_2\cdots\sin x_n}{\sin(x_1+x_2)\sin(x_2+x_3)\cdots\sin(x_n+x_1)}\le\frac{\sin^n(\pi/n)}{\sin^n(2\pi/n)}$, for $\sum x_i=\pi$

40 votes
5 answers
4k views

Stronger than Nesbitt inequality

  • 4,221
40 votes
8 answers
6k views

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $x^2y+y^2z+z^2x < \frac12$

  • 4,221
39 votes
2 answers
1k views

there exist infinite many $n\in\mathbb{N}$ such that $S_n-[S_n]<\frac{1}{n^2}$

38 votes
1 answer
2k views

$[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$

  • 7,368
37 votes
7 answers
3k views

Inequality $\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$

35 votes
10 answers
3k views

If both $a,b>0$, then $a^ab^b \ge a^bb^a$ [closed]

34 votes
2 answers
3k views

Trig sum: $\tan ^21^\circ+\tan ^22^\circ+\cdots+\tan^2 89^\circ = \text{?}$

  • 3,045
31 votes
2 answers
1k views

Prove $\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge 4+(x-y)^2$ for $4 \le x + y + z \le 5$

31 votes
10 answers
56k views

What is the probability that the center of the circle is contained within a triangle formed by choosing three random points on the circumference? [closed]

  • 2,425
30 votes
5 answers
8k views

If $A$ and $B$ are matrices such that $AB^2=BA$ and $A^4=I$ then find $B^{16}$.

29 votes
2 answers
1k views

Prove $\left(\frac{a+1}{a+b}\right)^a+\left(\frac{b+1}{b+c}\right)^b+\left(\frac{c+1}{c+a}\right)^c \geqslant 3$

  • 4,221
28 votes
6 answers
4k views

Importance of group action in abstract algebra [closed]

  • 1,494
28 votes
4 answers
5k views

Proving that the triangle inequality holds for a metric on $\mathbb{C}$

  • 12.1k
26 votes
7 answers
1k views

How to prove $\left(\frac{n}{n+1}\right)^{n+1}<\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}<\left(\frac{n}{n+1}\right)^n$

26 votes
2 answers
1k views

How to prove this inequality(7)?

25 votes
2 answers
3k views

Sum of derivatives of a polynomial

  • 505
24 votes
3 answers
773 views

$\triangle ABC$ with a point $D$ inside has $\angle BAD=114^\circ$, $\angle DAC=6^\circ$, $\angle ACD=12^\circ$, and $\angle DCB=18^\circ$.

  • 350
24 votes
3 answers
660 views

How prove this $\{a\}\cdot\{b\}\cdot\{c\}=0$ if $\lfloor na\rfloor+\lfloor nb\rfloor=\lfloor nc\rfloor$

24 votes
3 answers
2k views

Prove $\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$

  • 4,221
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