Question

If $a,b,c ∈ \mathbb R^+$ prove $(\frac{a}{b+c} + \frac{1}{2})(\frac{b}{c+a} + \frac{1}{2})(\frac{c}{a+b} + \frac{1}{2}) \ge 1$

score 116 · Answer 1 · 2023-10-01 09:23:25

116

Views

If $a,b,c ∈ \mathbb R^+$ prove $(\frac{a}{b+c} + \frac{1}{2})(\frac{b}{c+a} + \frac{1}{2})(\frac{c}{a+b} + \frac{1}{2}) \ge 1$

Published on 01 Oct 2023 - 9:23

score 111 · Answer 2 · 2023-10-02 19:25:51

111

Views

$a,b,c ∈ \mathbb R^+$ and $a+b=1$ then find max value of $a^4b + ab^4$

Published on 02 Oct 2023 - 19:25

score 100 · Answer 3 · 2023-10-06 14:10:49

100

Views

If $a,b,c > 0$ prove $\frac{(b+c-a)^2}{a}+\frac{(a+c-b)^2}{b}+\frac{(a+b-c)^2}{c}\ge\frac{b^2+c^2}{b+c}+\frac{a^2+c^2}{a+c}+\frac{a^2+b^2}{a+b}$

Published on 06 Oct 2023 - 14:10

score 49 · Answer 4 · 2026-03-26 09:32:29

49

Views

Seeking Clarification on an Inequality Involving Roots and Products

Published on 26 Mar 2026 - 9:32

score 109 · Answer 5 · 2026-03-26 07:35:23

109

Views

How to prove this inequality $\sum_{r=1}^{n}a_{r}\sqrt{\frac{n-1}{1-a_{r}}}\ge\sum_{r=1}^{n}\sqrt{a_{r}}$

Published on 26 Mar 2026 - 7:35

score 542 · Answer 6 · 2023-10-15 04:19:10

542

Views

Minimum value of expression $\displaystyle \sqrt{16b^4+(b-33)^2}$

Published on 15 Oct 2023 - 4:19

score 46 · Answer 7 · 2026-03-26 09:31:23

46

Views

Proving Inequality Involving Sums and Square Roots with Given Conditions

Published on 26 Mar 2026 - 9:31

score 136 · Answer 8 · 2026-03-29 11:44:28

136

Views

Let $a, b,c$ be positive real numbers such that $a+b+c=1,$ prove $(a + b) ^ 2 (1 + 2c)(2a + 3c)(2b + 3c) \ge54abc$

Published on 29 Mar 2026 - 11:44

score 159 · Answer 9 · 2026-03-26 09:30:42

159

Views

Brazilian Mathematical Olympiad 2023, Level U, Problem 3

Published on 26 Mar 2026 - 9:30

score 67 · Answer 10 · 2023-10-22 13:45:38

67

Views

Let $n \geq 3$ be an integer and $k>1$ be a real number.

Published on 22 Oct 2023 - 13:45

score 194 · Answer 11 · 2023-10-22 17:13:17

194

Views

If $a^2+ab+bc+ca<0$, show that $a^2<b^2+c^2$

Published on 22 Oct 2023 - 17:13

score 50 · Answer 12 · 2023-10-23 07:13:54

50

Views

Basic Inequality problem related to AM-GM Inequalities

Published on 23 Oct 2023 - 7:13

score 75 · Answer 13 · 2026-03-25 12:27:22

75

Views

Max $P= \frac{2ab+3b^2}{(a+3b)^2}+\frac{2bc+3c^2}{(b+3c)^2}+\frac{2ca+a^2}{(c+3a)^2}$

Published on 25 Mar 2026 - 12:27

score 47 · Answer 14 · 2023-11-04 08:09:18

47

Views

Minimum of $\sum_{cyc}\frac1x$, $\sum_{cyc}x=3a>0$, $x,y,z>0$

Published on 04 Nov 2023 - 8:09

score 41 · Answer 15 · 2026-03-27 15:18:17

41

Views

How to prove this integral inequality $\int_{ - 1}^1 {{{\left| {g\left( s \right)} \right|}^2}ds}$

Published on 27 Mar 2026 - 15:18

List Question

If $a,b,c ∈ \mathbb R^+$ prove $(\frac{a}{b+c} + \frac{1}{2})(\frac{b}{c+a} + \frac{1}{2})(\frac{c}{a+b} + \frac{1}{2}) \ge 1$

$a,b,c ∈ \mathbb R^+$ and $a+b=1$ then find max value of $a^4b + ab^4$

If $a,b,c > 0$ prove $\frac{(b+c-a)^2}{a}+\frac{(a+c-b)^2}{b}+\frac{(a+b-c)^2}{c}\ge\frac{b^2+c^2}{b+c}+\frac{a^2+c^2}{a+c}+\frac{a^2+b^2}{a+b}$

Seeking Clarification on an Inequality Involving Roots and Products

How to prove this inequality $\sum_{r=1}^{n}a_{r}\sqrt{\frac{n-1}{1-a_{r}}}\ge\sum_{r=1}^{n}\sqrt{a_{r}}$

Minimum value of expression $\displaystyle \sqrt{16b^4+(b-33)^2}$

Proving Inequality Involving Sums and Square Roots with Given Conditions

Let $a, b,c$ be positive real numbers such that $a+b+c=1,$ prove $(a + b) ^ 2 (1 + 2c)(2a + 3c)(2b + 3c) \ge54abc$

Brazilian Mathematical Olympiad 2023, Level U, Problem 3

Let $n \geq 3$ be an integer and $k>1$ be a real number.

If $a^2+ab+bc+ca<0$, show that $a^2<b^2+c^2$

Basic Inequality problem related to AM-GM Inequalities

Max $P= \frac{2ab+3b^2}{(a+3b)^2}+\frac{2bc+3c^2}{(b+3c)^2}+\frac{2ca+a^2}{(c+3a)^2}$

Minimum of $\sum_{cyc}\frac1x$, $\sum_{cyc}x=3a>0$, $x,y,z>0$

How to prove this integral inequality $\int_{ - 1}^1 {{{\left| {g\left( s \right)} \right|}^2}ds}$

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