Question

If $x + y + z = 2$, then show $\frac{x y z}{(1-x)(1-y)(1-z)} \geq 8$, added a second question(Problem 2).

score 105 · Answer 1 · 2026-03-27 01:04:18

105

Views

If $x + y + z = 2$, then show $\frac{x y z}{(1-x)(1-y)(1-z)} \geq 8$, added a second question(Problem 2).

Published on 27 Mar 2026 - 1:04

score 308 · Answer 2 · 2026-03-26 17:38:23

308

Views

Find Maxima and Minima of $f( \theta) = a \sin^2 \theta + b \sin \theta \cos \theta + c \cos^2 \theta$

Published on 26 Mar 2026 - 17:38

score 99 · Answer 3 · 2026-03-25 03:04:18

99

Views

Find the minimum value of $\sum_{cyc}\frac{1}{\sqrt{a^2 - ab + 3b^2 + 1}}$ where $a, b, c > 0$ and $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \le 3$.

Published on 25 Mar 2026 - 3:04

score 149 · Answer 4 · 2026-03-27 04:38:08

149

Views

Prove for all positive a,b,c that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$

Published on 27 Mar 2026 - 4:38

score 104 · Answer 5 · 2026-03-25 09:29:13

104

Views

Calculate the minimum value of $\sum_\mathrm{cyc}\frac{a^2}{b + c}$ where $a, b, c > 0$ and $\sum_\mathrm{cyc}\sqrt{a^2 + b^2} = 1$.

Published on 25 Mar 2026 - 9:29

score 244 · Answer 6 · 2026-03-24 23:41:50

244

Views

Minimize this real function on $\mathbb{R}^{2}$ without calculus?

Published on 24 Mar 2026 - 23:41

score 87 · Answer 7 · 2026-03-26 02:56:22

87

Views

Prove that $\sum_{cyc}\frac{1}{x^2 + 1} \ge \frac{2}{3}\bigl(\sum_{cyc}\frac{x}{\sqrt{x^2 + 1}}\bigr)^3$ where $x, y, z > 0$ and $xy + yz + zx = 1$.

Published on 26 Mar 2026 - 2:56

score 547 · Answer 8 · 2019-06-05 20:13:50

547

Views

Let n be a positive integer such that

Published on 05 Jun 2019 - 20:13

score 101 · Answer 9 · 2026-03-27 01:04:17

101

Views

Calculate the maximum value of $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$ where $3a + 4b + 5c = 12$

Published on 27 Mar 2026 - 1:04

score 158 · Answer 10 · 2026-03-26 02:56:21

158

Views

$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$ given that where $x, y, z > 0$ and $xyz = \frac{1}{2}$.

Published on 26 Mar 2026 - 2:56

score 654 · Answer 11 · 2019-06-09 18:32:15

654

Views

Maximum and minimum of $\cos^2x+\sin^2y$, where $x-y=\pi/4$ and $0\leq x\leq \pi $

Published on 09 Jun 2019 - 18:32

score 75 · Answer 12 · 2026-03-25 09:28:30

75

Views

Prove $\sum_{\mathrm{cyc}} (\frac{a^2}{3bc}+\frac{a(b+c)}{b^2+c^2})\ge 4$

Published on 25 Mar 2026 - 9:28

score 594 · Answer 13 · 2026-03-24 14:54:51

594

Views

Is $\left(1+\frac1n\right)^{n+1/2}$ decreasing?

Published on 24 Mar 2026 - 14:54

score 50 · Answer 14 · 2019-06-13 17:14:07

50

Views

Prove that $3 \le \sum_{cyc}a\sqrt{b^3 + 1} \le \sum_{cyc}ab^2 + 3$ where $a, b, c \ge 0$ and $a + b + c = 3$.

Published on 13 Jun 2019 - 17:14

score 1.5k · Answer 15 · 2026-03-27 07:47:02

1.5k

Views

Range of a function $|\sin x|+|\cos x|$

Published on 27 Mar 2026 - 7:47

List Question

If $x + y + z = 2$, then show $\frac{x y z}{(1-x)(1-y)(1-z)} \geq 8$, added a second question(Problem 2).

Find Maxima and Minima of $f( \theta) = a \sin^2 \theta + b \sin \theta \cos \theta + c \cos^2 \theta$

Find the minimum value of $\sum_{cyc}\frac{1}{\sqrt{a^2 - ab + 3b^2 + 1}}$ where $a, b, c > 0$ and $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \le 3$.

Prove for all positive a,b,c that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$

Calculate the minimum value of $\sum_\mathrm{cyc}\frac{a^2}{b + c}$ where $a, b, c > 0$ and $\sum_\mathrm{cyc}\sqrt{a^2 + b^2} = 1$.

Minimize this real function on $\mathbb{R}^{2}$ without calculus?

Prove that $\sum_{cyc}\frac{1}{x^2 + 1} \ge \frac{2}{3}\bigl(\sum_{cyc}\frac{x}{\sqrt{x^2 + 1}}\bigr)^3$ where $x, y, z > 0$ and $xy + yz + zx = 1$.

Let n be a positive integer such that

Calculate the maximum value of $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$ where $3a + 4b + 5c = 12$

$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$ given that where $x, y, z > 0$ and $xyz = \frac{1}{2}$.

Maximum and minimum of $\cos^2x+\sin^2y$, where $x-y=\pi/4$ and $0\leq x\leq \pi $

Prove $\sum_{\mathrm{cyc}} (\frac{a^2}{3bc}+\frac{a(b+c)}{b^2+c^2})\ge 4$

Is $\left(1+\frac1n\right)^{n+1/2}$ decreasing?

Prove that $3 \le \sum_{cyc}a\sqrt{b^3 + 1} \le \sum_{cyc}ab^2 + 3$ where $a, b, c \ge 0$ and $a + b + c = 3$.

Range of a function $|\sin x|+|\cos x|$

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