Question

Use the fact that $\ln(1+x) \leq x$ to show that $(a_1a_2\cdot\cdot\cdot a_n)^{1/n} \leq \frac{a_1+a_2+\cdot\cdot\cdot+a_n}{n}$

score 104 · Answer 1 · 2026-04-17 15:25:56

104

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Use the fact that $\ln(1+x) \leq x$ to show that $(a_1a_2\cdot\cdot\cdot a_n)^{1/n} \leq \frac{a_1+a_2+\cdot\cdot\cdot+a_n}{n}$

Published on 17 Apr 2026 - 15:25

score 154 · Answer 2 · 2026-05-10 14:37:55

154

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What tools one should use for inequalities?

Published on 10 May 2026 - 14:37

score 91 · Answer 3 · 2026-04-14 21:02:59

91

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Showing that $x+y+z \le 2\left(\frac {x^2}{y+z}+ \frac {y^2}{x+z}+ \frac {z^2}{x+y}\right)$ for positive $x, y, z$?

Published on 14 Apr 2026 - 21:02

score 127 · Answer 4 · 2026-04-16 13:22:46

127

Views

For $x\geq 0$, what is the smallest value of $\frac{4x^2+8x+13}{6(x+1)}$?

Published on 16 Apr 2026 - 13:22

score 158 · Answer 5 · 2026-04-11 21:21:40

158

Views

Using Method of Undetermined coefficients to find maximum value

Published on 11 Apr 2026 - 21:21

score 246 · Answer 6 · 2026-04-15 23:33:22

246

Views

Intermediate inequalities; is there a way to know if you're getting a "bad deal"?

Published on 15 Apr 2026 - 23:33

score 107 · Answer 7 · 2026-05-10 14:18:01

107

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An inequality with substitution

Published on 10 May 2026 - 14:18

score 181 · Answer 8 · 2026-04-17 14:33:42

181

Views

Prove: $\log_2(x)+\log_x(y)+\log_y(8)\geq \sqrt[3]{81}$

Published on 17 Apr 2026 - 14:33

score 172 · Answer 9 · 2026-04-15 23:00:46

172

Views

Show $\frac{1}{b+c+d} + \frac{1}{a+c+d} + \frac{1}{a+b+d} + \frac{1}{a+b+c} \ge \frac{16}{3(a+b+c+d)}$.

Published on 15 Apr 2026 - 23:00

score 78 · Answer 10 · 2026-04-09 13:51:04

78

Views

How do I prove the following inequality?

Published on 09 Apr 2026 - 13:51

score 263 · Answer 11 · 2026-05-10 14:40:35

263

Views

$ \frac {3 x y z}{(x+y) (y+z) (z+x)} + \sum\limits_{cycl}^{} \left(\frac {x+y}{x+y+ 2 z}\right)^2 \ge \frac {9}{8}.$

Published on 10 May 2026 - 14:40

score 209 · Answer 12 · 2026-04-17 02:58:06

209

Views

Showing that if $p_1 + \cdots p_n = 1$ then $\displaystyle \sum_{k=1}^n \left(p_k + \dfrac {1}{p_k} \right)^2 \ge n^3+2n+\dfrac 1n$?

Published on 17 Apr 2026 - 2:58

score 245 · Answer 13 · 2026-04-17 13:41:59

245

Views

Prove: $\log_2(x)+\log_3(x)+\log_5(x)>9\log_{30}(x)$

Published on 17 Apr 2026 - 13:41

score 415 · Answer 14 · 2026-04-16 00:34:14

415

Views

Prove that $a^4+b^4+1\ge a+b$.

Published on 16 Apr 2026 - 0:34

score 105 · Answer 15 · 2026-04-15 18:49:33

105

Views

Algebraic Inequality involving AM-GM-HM

Published on 15 Apr 2026 - 18:49

List Question

Use the fact that $\ln(1+x) \leq x$ to show that $(a_1a_2\cdot\cdot\cdot a_n)^{1/n} \leq \frac{a_1+a_2+\cdot\cdot\cdot+a_n}{n}$

What tools one should use for inequalities?

Showing that $x+y+z \le 2\left(\frac {x^2}{y+z}+ \frac {y^2}{x+z}+ \frac {z^2}{x+y}\right)$ for positive $x, y, z$?

For $x\geq 0$, what is the smallest value of $\frac{4x^2+8x+13}{6(x+1)}$?

Using Method of Undetermined coefficients to find maximum value

Intermediate inequalities; is there a way to know if you're getting a "bad deal"?

An inequality with substitution

Prove: $\log_2(x)+\log_x(y)+\log_y(8)\geq \sqrt[3]{81}$

Show $\frac{1}{b+c+d} + \frac{1}{a+c+d} + \frac{1}{a+b+d} + \frac{1}{a+b+c} \ge \frac{16}{3(a+b+c+d)}$.

How do I prove the following inequality?

$ \frac {3 x y z}{(x+y) (y+z) (z+x)} + \sum\limits_{cycl}^{} \left(\frac {x+y}{x+y+ 2 z}\right)^2 \ge \frac {9}{8}.$

Showing that if $p_1 + \cdots p_n = 1$ then $\displaystyle \sum_{k=1}^n \left(p_k + \dfrac {1}{p_k} \right)^2 \ge n^3+2n+\dfrac 1n$?

Prove: $\log_2(x)+\log_3(x)+\log_5(x)>9\log_{30}(x)$

Prove that $a^4+b^4+1\ge a+b$.

Algebraic Inequality involving AM-GM-HM

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