Michael Rozenberg's user avatar
Michael Rozenberg's user avatar
Michael Rozenberg's user avatar
Michael Rozenberg
  • Member for 7 years, 11 months
  • Last seen this week
  • Tel-Aviv
85 votes
7 answers
10k views

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

44 votes
2 answers
3k views

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

37 votes
7 answers
3k views

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

29 votes
1 answer
1k views

Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$

26 votes
1 answer
8k views

Finding all $ f : \mathbb R \to \mathbb R $ satisfying $ f \bigl ( f ( x ) f ( y ) \bigr ) + f ( x + y ) = f ( x y ) $ for all $ x , y \in \mathbb R $

25 votes
4 answers
2k views

Inequality with five variables

23 votes
5 answers
1k views

If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$

22 votes
2 answers
443 views

Prove that $\sum\limits_{cyc}\frac{a}{\sqrt{a+3b}}\geq\sqrt{a+b+c+d}$

18 votes
1 answer
773 views

Geometric inequality $\frac{R_a}{2a+b}+\frac{R_b}{2b+c}+\frac{R_c}{2c+a}\geq\frac{1}{\sqrt3}$

18 votes
2 answers
915 views

If $\{x,y,z\}\subset[-1,1]$ and $x+y+z=0$ so $\sum\limits_{cyc}\sqrt{1+x+\frac{y^2}{6}}\leq3$

17 votes
5 answers
2k views

Prove that $\sum\limits_{cyc}\frac{a}{a^{11}+1}\leq\frac{3}{2}$ for $a, b, c > 0$ with $abc = 1$

16 votes
1 answer
554 views

If $\{a,b,c,d,e\}\subset[0,1]$ so $\sum\limits_{cyc}\frac{1}{1+a+b}\leq\frac{5}{1+2\sqrt[5]{abcde}}$

14 votes
2 answers
263 views

If $a+b+c+d=1$ so $\sum\limits_{cyc}\sqrt{a+b+c^2}\geq3$

14 votes
3 answers
8k views

If $a^3+b^3+c^3=3$ so $\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a}\geq\frac{3}{2}$

14 votes
3 answers
522 views

If $a+b+c=1$ then $\sum\limits_{cyc}\frac{a}{\sqrt[3]{a+b}}\leq\frac{31}{27}$

12 votes
1 answer
487 views

Prove that $\sum\limits_{cyc}\sqrt[3]{a^2+4bc}\geq\sqrt[3]{45(ab+ac+bc)}$

12 votes
3 answers
2k views

The number of ways to represent a natural number as the sum of three different natural numbers

12 votes
3 answers
368 views

An application of the Casey's theorem.

12 votes
2 answers
608 views

Prove that if $x_{n+2}=\frac{2+x_{n+1}}{2+x_n},$ then $x_n$ converges

11 votes
2 answers
361 views

If $abc=1$ so $\sum\limits_{cyc}\sqrt{\frac{a}{4a+2b+3}}\leq1$.

11 votes
2 answers
283 views

Prove that $\sum\limits_{cyc}\frac{a}{a+b}\geq1+\frac{3\sqrt[3]{a^2b^2c^2}}{2(ab+ac+bc)}$

10 votes
2 answers
446 views

If $xy+xz+yz=1+2xyz$ then $\sqrt{x}+\sqrt{y}+\sqrt{z}\geq2$.

10 votes
8 answers
739 views

Prove that $\ln2<\frac{1}{\sqrt[3]3}$

10 votes
4 answers
437 views

Prove that $\sum\limits_{cyc}\frac{a^2}{a^3+2}\leq\frac{4}{3}$ if $a, b, c, d > 0$ and $abcd=1$

10 votes
1 answer
520 views

Very strong inequality

10 votes
1 answer
253 views

Without calculator prove that $\log^211+\log^29<\log99$

9 votes
0 answers
150 views

Solving the system $\tan x+\sin y+\sin z=3x$, $\sin x+\tan y+\sin z=3y$, $\sin x+\sin y+\tan z=3z$

9 votes
2 answers
305 views

Prove that $\frac{a}{1+a^2}+\frac{b}{1+a^2+b^2}+\frac{c}{1+a^2+b^2+c^2}+\frac{d}{1+a^2+b^2+c^2+d^2}\leq\frac{3}{2}$

8 votes
1 answer
150 views

Given $\sum\limits_{i=1}^6a_i^2=6$, where $a_i>0$, $a_7=a_1$. Prove that $\sum\limits_{i=1}^6\frac{a_i^2}{a_{i+1}}\geq6$

7 votes
2 answers
158 views

If $a+b+c+d=0$ and $\{a,b,c,d\}\subset[-1,1]$ so $\sum\limits_{cyc}\sqrt{1+a+b^2}\geq4$