Question

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

score 64 · Answer 1 · 2026-04-13 08:44:25

64

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

Published on 13 Apr 2026 - 8:44

score 77 · Answer 2 · 2026-05-10 17:17:38

77

Views

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

Published on 10 May 2026 - 17:17

score 580 · Answer 3 · 2026-04-16 06:12:53

580

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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 16 Apr 2026 - 6:12

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 99 · Answer 5 · 2026-04-13 06:26:00

99

Views

Prove that in any triangle

Published on 13 Apr 2026 - 6:26

score 97 · Answer 6 · 2026-04-15 18:03:23

97

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 15 Apr 2026 - 18:03

score 789 · Answer 7 · 2026-04-13 05:47:54

789

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 13 Apr 2026 - 5:47

score 132 · Answer 8 · 2026-04-16 22:42:42

132

Views

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

Published on 16 Apr 2026 - 22:42

score 436 · Answer 9 · 2026-04-16 13:11:05

436

Views

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

Published on 16 Apr 2026 - 13:11

score 48 · Answer 10 · 2026-04-15 13:55:59

48

Views

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

Published on 15 Apr 2026 - 13:55

score 242 · Answer 11 · 2026-04-11 22:22:37

242

Views

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

Published on 11 Apr 2026 - 22:22

score 899 · Answer 12 · 2026-04-18 10:11:02

899

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 18 Apr 2026 - 10:11

score 106 · Answer 13 · 2026-04-11 15:15:55

106

Views

Proof of inequality in triangle

Published on 11 Apr 2026 - 15:15

score 92 · Answer 14 · 2026-05-10 17:15:44

92

Views

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

Published on 10 May 2026 - 17:15

score 151 · Answer 15 · 2026-04-17 00:47:39

151

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 17 Apr 2026 - 0:47

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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