Question

Proof and Geometric intuition of $u \leq 2\ln(1+u)$ for $u \in [0,1]$.

score 113 · Answer 1 · 2026-05-10 14:49:28

113

Views

Proof and Geometric intuition of $u \leq 2\ln(1+u)$ for $u \in [0,1]$.

Published on 10 May 2026 - 14:49

score 127 · Answer 2 · 2026-04-15 12:58:25

127

Views

What is wrong with my solution of maximum value of $ \sin \frac {A}{2} + \sin \frac{B}{2} + \sin \frac{C}{2} $ in a triangle ABC?

Published on 15 Apr 2026 - 12:58

score 151 · Answer 3 · 2026-05-10 15:00:16

151

Views

Prove $a^2b(a-b)+ b^2c(b-c)+ c^2a(c-a)\geq 0$

Published on 10 May 2026 - 15:00

score 345 · Answer 4 · 2026-05-10 15:01:31

345

Views

Understanding a Proof: The square root of any metric is ptolemaic..

Published on 10 May 2026 - 15:01

score 586 · Answer 5 · 2026-04-16 04:29:11

586

Views

Angle formed by orthocenter, incenter and circumcenter of a triangle $>135^\circ$?

Published on 16 Apr 2026 - 4:29

score 245 · Answer 6 · 2026-05-10 14:32:38

245

Views

Let $A, B$ and $C$ be the angles of an acute triangle. Show that: $\sin A+\sin B +\sin C > 2$.

Published on 10 May 2026 - 14:32

score 33 · Answer 7 · 2026-04-16 08:45:00

33

Views

For acute triangle with sides $a$, $b$, $c$ and circumradius $R$, define $x$, $y$, $z$ with $a+x=b+y=c+z=2R$. Show $x+y+z\lt2R$.

Published on 16 Apr 2026 - 8:45

score 31 · Answer 8 · 2026-04-13 19:55:19

31

Views

$a+OA\lt b+OB\lt c+OC$ when $a\lt b\lt c$ in a triangle

Published on 13 Apr 2026 - 19:55

score 581 · Answer 9 · 2026-04-16 11:51:30

581

Views

If $p$, $q$, $r$ and $s$ are four sides of a quadrilateral, then find the minimum value of $\frac{p^2+ q^2 + r^2}{s^2}$ with logic

Published on 16 Apr 2026 - 11:51

score 170 · Answer 10 · 2026-04-16 05:38:24

170

Views

Find the minimum value of the perimeter

Published on 16 Apr 2026 - 5:38

score 49 · Answer 11 · 2026-04-16 19:15:44

49

Views

$\sqrt{5}BA \leq PA +PB+\sqrt{2}PC$

Published on 16 Apr 2026 - 19:15

score 195 · Answer 12 · 2026-05-10 19:25:26

195

Views

Given three positive numbers $a,b,c$ so that $abc= 1$. Prove $(a-1+\frac{1}{b})(b-1+\frac{1}{c})(c-1+\frac{1}{a})\leqq\frac{2}{a+b+c-1}$ .

Published on 10 May 2026 - 19:25

score 60 · Answer 13 · 2026-04-16 05:41:47

60

Views

Can we conclude $0^\circ<A\leq B\leq C<180^\circ$ for any $\triangle ABC$ with $0<a\leq b\leq c$?

Published on 16 Apr 2026 - 5:41

score 330 · Answer 14 · 2026-05-10 19:25:18

330

Views

Given three positive numbers $x,y,z$, prove that $(xyz+x^{2}y+y^{2}z+z^{2}x)^{4}\geqq\frac{256}{27}(x+y+z)^{3}x^{3}y^{3}z^{3}$ .

Published on 10 May 2026 - 19:25

score 77 · Answer 15 · 2026-04-15 07:58:24

77

Views

Let $a,b,c$ be the sides of a triangle. The find maximum value $ \frac{A}{S^{2}}$

Published on 15 Apr 2026 - 7:58

List Question

Proof and Geometric intuition of $u \leq 2\ln(1+u)$ for $u \in [0,1]$.

What is wrong with my solution of maximum value of $ \sin \frac {A}{2} + \sin \frac{B}{2} + \sin \frac{C}{2} $ in a triangle ABC?

Prove $a^2b(a-b)+ b^2c(b-c)+ c^2a(c-a)\geq 0$

Understanding a Proof: The square root of any metric is ptolemaic..

Angle formed by orthocenter, incenter and circumcenter of a triangle $>135^\circ$?

Let $A, B$ and $C$ be the angles of an acute triangle. Show that: $\sin A+\sin B +\sin C > 2$.

For acute triangle with sides $a$, $b$, $c$ and circumradius $R$, define $x$, $y$, $z$ with $a+x=b+y=c+z=2R$. Show $x+y+z\lt2R$.

$a+OA\lt b+OB\lt c+OC$ when $a\lt b\lt c$ in a triangle

If $p$, $q$, $r$ and $s$ are four sides of a quadrilateral, then find the minimum value of $\frac{p^2+ q^2 + r^2}{s^2}$ with logic

Find the minimum value of the perimeter

$\sqrt{5}BA \leq PA +PB+\sqrt{2}PC$

Given three positive numbers $a,b,c$ so that $abc= 1$. Prove $(a-1+\frac{1}{b})(b-1+\frac{1}{c})(c-1+\frac{1}{a})\leqq\frac{2}{a+b+c-1}$ .

Can we conclude $0^\circ<A\leq B\leq C<180^\circ$ for any $\triangle ABC$ with $0<a\leq b\leq c$?

Given three positive numbers $x,y,z$, prove that $(xyz+x^{2}y+y^{2}z+z^{2}x)^{4}\geqq\frac{256}{27}(x+y+z)^{3}x^{3}y^{3}z^{3}$ .

Let $a,b,c$ be the sides of a triangle. The find maximum value $ \frac{A}{S^{2}}$

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