Question

Minimal area of a right triangle with inradius $1$

score 504 · Answer 1 · 2018-06-30 08:23:47

504

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Minimal area of a right triangle with inradius $1$

Published on 30 Jun 2018 - 8:23

score 123 · Answer 2 · 2018-07-01 11:06:45

123

Views

Let $a,b,c$ be the length of the side $BC,CA,AB$ respectively for $\triangle ABC$. Show that $(2b+2c-2a)^3(a+b+c)\geq18a^2bc$.

Published on 01 Jul 2018 - 11:06

score 70 · Answer 3 · 2018-07-15 15:13:58

70

Views

Draw a graph for the inequality $(x_1 - \frac{1}{2})^2 + (y_1 - \frac{1}{2})^2 < (x_2 - \frac{1}{2})^2 + (y_2 - \frac{1}{2})^2$

Published on 15 Jul 2018 - 15:13

score 68 · Answer 4 · 2026-03-25 07:38:56

68

Views

inequality proving for intersection

Published on 25 Mar 2026 - 7:38

score 196 · Answer 5 · 2018-07-25 20:11:50

196

Views

Proving $\frac{1}{\sin(A/2)}+\frac{1}{\sin(B/2)}+\frac{1}{\sin(C/2)}\ge 6$, where $A$, $B$, $C$ are angles of a triangle

Published on 25 Jul 2018 - 20:11

score 92 · Answer 6 · 2018-08-05 15:42:12

92

Views

Prove inequality inside a triangle

Published on 05 Aug 2018 - 15:42

score 344 · Answer 7 · 2018-08-06 19:21:11

344

Views

Prove that $\frac{6(a^2 + b^2 + c^2)}{a + b + c} \geq \frac{(a + b)^2}{b + c} + \frac{(b + c)^2}{c + a} + \frac{(a + c)^2}{a + b}$

Published on 06 Aug 2018 - 19:21

score 249 · Answer 8 · 2018-08-10 04:14:54

249

Views

inequality in geometry

Published on 10 Aug 2018 - 4:14

score 114 · Answer 9 · 2026-03-25 12:52:37

114

Views

Inequality about Schur

Published on 25 Mar 2026 - 12:52

score 54 · Answer 10 · 2018-08-28 02:53:20

54

Views

$A_1, A_2\in [AB]$ s.t. $AA_1 < AA_2 < 1$

Published on 28 Aug 2018 - 2:53

score 115 · Answer 11 · 2018-09-03 03:12:29

115

Views

Finding the values of $x$ and $y$ for which $x^2 + y^2$ is a minimum

Published on 03 Sep 2018 - 3:12

score 177 · Answer 12 · 2018-09-06 22:09:20

177

Views

Permutation of points $P_i\in X$ such that $\sum^n_{j=1}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8$

Published on 06 Sep 2018 - 22:09

score 127 · Answer 13 · 2026-03-25 01:40:04

127

Views

Show that $\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$ for a triangle with sides $a$, $b$, $c$ and area $S$

Published on 25 Mar 2026 - 1:40

score 96 · Answer 14 · 2018-09-21 07:11:41

96

Views

Given triangle sides $a$, $b$, $c$, show $\sum_{\text{cyc}}\frac{a}{a+b}\geq\frac32\prod_{\text{cyc}}(\frac{a}{a+b}+\frac{b}{b+c})$

Published on 21 Sep 2018 - 7:11

score 46 · Answer 15 · 2018-09-22 09:23:37

46

Views

lower and upper bounds on $aA+ bB + cC$ and ${\large{\frac{ab}{l_c}}}+{\large{\frac{bc}{l_a}}}+ {\large{\frac{ca}{l_b}}}$.

Published on 22 Sep 2018 - 9:23

List Question

Minimal area of a right triangle with inradius $1$

Let $a,b,c$ be the length of the side $BC,CA,AB$ respectively for $\triangle ABC$. Show that $(2b+2c-2a)^3(a+b+c)\geq18a^2bc$.

Draw a graph for the inequality $(x_1 - \frac{1}{2})^2 + (y_1 - \frac{1}{2})^2 < (x_2 - \frac{1}{2})^2 + (y_2 - \frac{1}{2})^2$

inequality proving for intersection

Proving $\frac{1}{\sin(A/2)}+\frac{1}{\sin(B/2)}+\frac{1}{\sin(C/2)}\ge 6$, where $A$, $B$, $C$ are angles of a triangle

Prove inequality inside a triangle

Prove that $\frac{6(a^2 + b^2 + c^2)}{a + b + c} \geq \frac{(a + b)^2}{b + c} + \frac{(b + c)^2}{c + a} + \frac{(a + c)^2}{a + b}$

inequality in geometry

Inequality about Schur

$A_1, A_2\in [AB]$ s.t. $AA_1 < AA_2 < 1$

Finding the values of $x$ and $y$ for which $x^2 + y^2$ is a minimum

Permutation of points $P_i\in X$ such that $\sum^n_{j=1}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8$

Show that $\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$ for a triangle with sides $a$, $b$, $c$ and area $S$

Given triangle sides $a$, $b$, $c$, show $\sum_{\text{cyc}}\frac{a}{a+b}\geq\frac32\prod_{\text{cyc}}(\frac{a}{a+b}+\frac{b}{b+c})$

lower and upper bounds on $aA+ bB + cC$ and ${\large{\frac{ab}{l_c}}}+{\large{\frac{bc}{l_a}}}+ {\large{\frac{ca}{l_b}}}$.

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