Question

Determining whether three values can be the lengths of a triangle or not

score 61 · Answer 1 · 2019-10-18 17:09:31

61

Views

Determining whether three values can be the lengths of a triangle or not

Published on 18 Oct 2019 - 17:09

score 76 · Answer 2 · 2026-03-25 11:04:31

76

Views

Prove that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ $\geq$ $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Published on 25 Mar 2026 - 11:04

score 578 · Answer 3 · 2019-11-15 19:39:12

578

Views

Find the smallest value of the following expression $\sqrt{(x-9)^2 +4} + \sqrt{x^2+y^2}+ \sqrt{(y-3)^2 +9}$

Published on 15 Nov 2019 - 19:39

score 146 · Answer 4 · 2026-03-25 21:54:08

146

Views

Show that $\frac{1}{MA^2}+\frac{1}{MB^2}+\frac{1}{MC^2}+\frac{1}{MD^2}\geq2$ for $M$ inside square $ABCD$ with side length $2$

Published on 25 Mar 2026 - 21:54

score 96 · Answer 5 · 2019-11-26 20:22:31

96

Views

Prove that in any triangle

Published on 26 Nov 2019 - 20:22

score 94 · Answer 6 · 2019-11-27 09:19:35

94

Views

If $\triangle$ is the area of triangle with side lengths $a,b,c$, then show that $\triangle \le\dfrac{1}{4}\cdot\sqrt{(a+b+c)abc}$

Published on 27 Nov 2019 - 9:19

score 786 · Answer 7 · 2019-11-29 21:32:03

786

Views

A bound for $\sqrt\frac{b+c-a}a+\sqrt\frac{c+a-b}b+\sqrt\frac{a+b-c}c$ in a triangle

Published on 29 Nov 2019 - 21:32

score 129 · Answer 8 · 2019-12-01 11:49:33

129

Views

Prove that $\frac{1}{q}+\frac{1}{p}>\frac{1}{r}$ when you know that:

Published on 01 Dec 2019 - 11:49

score 433 · Answer 9 · 2019-12-08 16:33:32

433

Views

prove that : $a^{2}+b^{2}+c^{2}+4abc<\frac12$

Published on 08 Dec 2019 - 16:33

score 45 · Answer 10 · 2019-12-16 17:25:50

45

Views

Given that R is interior to triangle BAD, prove $BR+DR<BA+AD$

Published on 16 Dec 2019 - 17:25

score 239 · Answer 11 · 2019-12-16 17:49:45

239

Views

In triangle $ABC$, find maximum value of $\sin A \cos B + \sin B \cos C + \sin C \cos A$

Published on 16 Dec 2019 - 17:49

score 897 · Answer 12 · 2019-12-23 16:52:06

897

Views

For regular tetrahedron $ABCD$ with center $O$, and $\overrightarrow{NO}=-3\overrightarrow{MO}$, is $NA+NB+NC+ND\geq MA+MB+MC+MD$?

Published on 23 Dec 2019 - 16:52

score 103 · Answer 13 · 2019-12-29 15:44:47

103

Views

Proof of inequality in triangle

Published on 29 Dec 2019 - 15:44

score 91 · Answer 14 · 2026-03-25 11:06:31

91

Views

Prove that for all acute triangles $\triangle ABC$, $r_a + r_b + r_c \ge m_a + m_b + m_c$.

Published on 25 Mar 2026 - 11:06

score 148 · Answer 15 · 2020-01-08 11:39:06

148

Views

Given $a, b, c > 0$ such that $a^2 + b^2 + c^2 + abc = 4$, prove that $\sum_{cyc}\frac{b}{\sqrt{(c^2 + 2)(a^2 + 2)}} \ge 1$.

Published on 08 Jan 2020 - 11:39

List Question

Determining whether three values can be the lengths of a triangle or not

Prove that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ $\geq$ $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Find the smallest value of the following expression $\sqrt{(x-9)^2 +4} + \sqrt{x^2+y^2}+ \sqrt{(y-3)^2 +9}$

Show that $\frac{1}{MA^2}+\frac{1}{MB^2}+\frac{1}{MC^2}+\frac{1}{MD^2}\geq2$ for $M$ inside square $ABCD$ with side length $2$

Prove that in any triangle

If $\triangle$ is the area of triangle with side lengths $a,b,c$, then show that $\triangle \le\dfrac{1}{4}\cdot\sqrt{(a+b+c)abc}$

A bound for $\sqrt\frac{b+c-a}a+\sqrt\frac{c+a-b}b+\sqrt\frac{a+b-c}c$ in a triangle

Prove that $\frac{1}{q}+\frac{1}{p}>\frac{1}{r}$ when you know that:

prove that : $a^{2}+b^{2}+c^{2}+4abc<\frac12$

Given that R is interior to triangle BAD, prove $BR+DR<BA+AD$

In triangle $ABC$, find maximum value of $\sin A \cos B + \sin B \cos C + \sin C \cos A$

For regular tetrahedron $ABCD$ with center $O$, and $\overrightarrow{NO}=-3\overrightarrow{MO}$, is $NA+NB+NC+ND\geq MA+MB+MC+MD$?

Proof of inequality in triangle

Prove that for all acute triangles $\triangle ABC$, $r_a + r_b + r_c \ge m_a + m_b + m_c$.

Given $a, b, c > 0$ such that $a^2 + b^2 + c^2 + abc = 4$, prove that $\sum_{cyc}\frac{b}{\sqrt{(c^2 + 2)(a^2 + 2)}} \ge 1$.

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