Question

Find the maximum value of $x^3 + y^3 + z^3$ where $x, y, z \in [0, 2]$ and $x + y + z = 3$.

score 155 · Answer 1 · 2026-03-25 03:22:13

155

Views

Find the maximum value of $x^3 + y^3 + z^3$ where $x, y, z \in [0, 2]$ and $x + y + z = 3$.

Published on 25 Mar 2026 - 3:22

score 153 · Answer 2 · 2026-03-25 03:21:23

153

Views

Number of real roots of $3^x+4^x=2^x+5^x$ with proof

Published on 25 Mar 2026 - 3:21

score 142 · Answer 3 · 2026-04-01 05:49:31

142

Views

Stronger than Jensen's inequality

Published on 01 Apr 2026 - 5:49

score 68 · Answer 4 · 2026-03-25 03:18:15

68

Views

Without majorization how to prove this inequality?

Published on 25 Mar 2026 - 3:18

score 49 · Answer 5 · 2026-03-25 03:19:42

49

Views

Is there a way to prove the next inequality without using the binomial series?

Published on 25 Mar 2026 - 3:19

score 412 · Answer 6 · 2026-05-10 12:12:22

412

Views

Upper Bound for a Sum

Published on 10 May 2026 - 12:12

score 77 · Answer 7 · 2026-03-25 03:22:12

77

Views

Prove that $(\sum_{i=1}^m (x_{i})^{p})^{q}\ge(\sum_{i=1}^m(x_{i})^{q})^{p}$ for all $x_i\in\mathbb R^+$ and $p,q\in\mathbb R^+$ such that $p \le q$

Published on 25 Mar 2026 - 3:22

score 194 · Answer 8 · 2026-03-25 03:22:14

194

Views

Inequality $\frac{\ln(7a+b)}{7a+b}+\frac{\ln(7b+c)}{7b+c}+\frac{\ln(7c+a)}{7c+a}\leq \frac{3\ln(8\sqrt{3})}{8\sqrt{3}}$

Published on 25 Mar 2026 - 3:22

score 119 · Answer 9 · 2026-03-25 03:19:43

119

Views

Range of function $f(x)=\sqrt{4-x^2}+\sqrt{x^2-1}$

Published on 25 Mar 2026 - 3:19

score 936 · Answer 10 · 2026-03-25 03:19:20

936

Views

Minimum and maximum sum of squares given constraints

Published on 25 Mar 2026 - 3:19

score 148 · Answer 11 · 2026-03-25 03:22:12

148

Views

Prove that $\sqrt{x^{3} + y^{3}+1} + \sqrt{z^{3} + y^{3}+1} + \sqrt{x^{3} + z^{3}+1} \ge 2 + \sqrt{2(x^{3} + y^{3}+z^{3})+1}$

Published on 25 Mar 2026 - 3:22

score 123 · Answer 12 · 2026-03-25 03:22:11

123

Views

Given three non-negative numbers $x,y,z$ so that $x+y+z=100$ and $x,y,z\notin (\!33,34\!)$. Prove that $xyz\leqq 33\cdot 34\cdot 33$ .

Published on 25 Mar 2026 - 3:22

score 174 · Answer 13 · 2026-03-25 03:22:13

174

Views

Prove that positive $x,y,z$ satisfy $\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$

Published on 25 Mar 2026 - 3:22

score 43 · Answer 14 · 2026-05-10 14:28:19

43

Views

Inequality with variable expoent

Published on 10 May 2026 - 14:28

score 105 · Answer 15 · 2026-05-04 09:02:45

105

Views

Prove that $\sum_{i=1}^n\frac{\sum_{i'=1}^na_{i'}^p - a_i^p}{\sum_{i'=1}^na_{i'}^q - a_i^q}\le n\cdot\frac{\sum_{i=1}^na_i^p}{\sum_{i=1}^na_i^q}$.

Published on 04 May 2026 - 9:02

List Question

Find the maximum value of $x^3 + y^3 + z^3$ where $x, y, z \in [0, 2]$ and $x + y + z = 3$.

Number of real roots of $3^x+4^x=2^x+5^x$ with proof

Stronger than Jensen's inequality

Without majorization how to prove this inequality?

Is there a way to prove the next inequality without using the binomial series?

Upper Bound for a Sum

Prove that $(\sum_{i=1}^m (x_{i})^{p})^{q}\ge(\sum_{i=1}^m(x_{i})^{q})^{p}$ for all $x_i\in\mathbb R^+$ and $p,q\in\mathbb R^+$ such that $p \le q$

Inequality $\frac{\ln(7a+b)}{7a+b}+\frac{\ln(7b+c)}{7b+c}+\frac{\ln(7c+a)}{7c+a}\leq \frac{3\ln(8\sqrt{3})}{8\sqrt{3}}$

Range of function $f(x)=\sqrt{4-x^2}+\sqrt{x^2-1}$

Minimum and maximum sum of squares given constraints

Prove that $\sqrt{x^{3} + y^{3}+1} + \sqrt{z^{3} + y^{3}+1} + \sqrt{x^{3} + z^{3}+1} \ge 2 + \sqrt{2(x^{3} + y^{3}+z^{3})+1}$

Given three non-negative numbers $x,y,z$ so that $x+y+z=100$ and $x,y,z\notin (\!33,34\!)$. Prove that $xyz\leqq 33\cdot 34\cdot 33$ .

Prove that positive $x,y,z$ satisfy $\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$

Inequality with variable expoent

Prove that $\sum_{i=1}^n\frac{\sum_{i'=1}^na_{i'}^p - a_i^p}{\sum_{i'=1}^na_{i'}^q - a_i^q}\le n\cdot\frac{\sum_{i=1}^na_i^p}{\sum_{i=1}^na_i^q}$.

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