Question

Find the minimum value of $\frac{a+b}{2} + \frac{2}{ab-b^{2}}$, with $a,b \in \mathbb{R}$, $a>b>0$

score 63 · Answer 1 · 2018-04-03 13:03:23

63

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Find the minimum value of $\frac{a+b}{2} + \frac{2}{ab-b^{2}}$, with $a,b \in \mathbb{R}$, $a>b>0$

Published on 03 Apr 2018 - 13:03

score 380 · Answer 2 · 2018-04-03 19:40:32

380

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A problem from XIV GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN The correspondence round, 2018

Published on 03 Apr 2018 - 19:40

score 274 · Answer 3 · 2026-03-25 22:11:13

274

Views

Diophantine Equation Concerning Powers of 2

Published on 25 Mar 2026 - 22:11

score 234 · Answer 4 · 2026-03-25 14:20:03

234

Views

Interesting 3 Variable Inequality for Real Numbers

Published on 25 Mar 2026 - 14:20

score 560 · Answer 5 · 2026-03-25 13:58:09

560

Views

Math Olympiad exercise $\frac{x}{\sqrt{3x+2y+z}}+\frac{y}{\sqrt{3y+2z+x}}+\frac{z}{\sqrt{3z+2x+y}}\leq\sqrt{\frac{x+y+z}{2}}$

Published on 25 Mar 2026 - 13:58

score 187 · Answer 6 · 2018-04-07 02:46:28

187

Views

Finding unit digit of $f(10)$

Published on 07 Apr 2018 - 2:46

score 239 · Answer 7 · 2026-03-25 00:01:37

239

Views

Symmetric inequality with three variables including radicals

Published on 25 Mar 2026 - 0:01

score 264 · Answer 8 · 2026-03-25 01:31:12

264

Views

Number of ways to gain $10$ points while gaining conditional $1$, $2$, or $3$ points at a time?

Published on 25 Mar 2026 - 1:31

score 566 · Answer 9 · 2018-04-08 16:12:13

566

Views

Number of $6$-digit numbers made up of the digits $1$, $2$, and $3$ with no digit occurring $3$ or more times consecutively?

Published on 08 Apr 2018 - 16:12

score 114 · Answer 10 · 2018-04-09 05:10:57

114

Views

Proving an Inequality - Functions

Published on 09 Apr 2018 - 5:10

score 220 · Answer 11 · 2018-04-09 12:18:36

Finding the minimum of $λ$ such that $|a-b|^p\le λ(2-|a-b|)\left|\frac{a|a|^{p-1}}{1-|a|}-\frac{b|b|^{p-1}}{1-|b|}\right|$ for $a,b\in(-1,1)$

score 332 · Answer 12 · 2018-04-10 21:48:31

332

Views

Prove: $(a+b) \sqrt {ab}+(a+c) \sqrt {ac}+(b+c)\sqrt {bc}≥ \frac {(a+b+c)^2}{2}$

Published on 10 Apr 2018 - 21:48

score 512 · Answer 13 · 2026-03-24 23:49:52

512

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Writing integers as a product of as few elements of $\{\frac21, \frac32, \frac43, \frac54, \ldots\}$ as possible

Published on 24 Mar 2026 - 23:49

score 187 · Answer 14 · 2026-03-25 18:04:36

187

Views

Converse of theorem of two tangent circles

Published on 25 Mar 2026 - 18:04

score 96 · Answer 15 · 2026-03-25 06:07:00

96

Views

Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $ f(x) \geq f(y)\sin(x-y)$

Published on 25 Mar 2026 - 6:07

List Question

Find the minimum value of $\frac{a+b}{2} + \frac{2}{ab-b^{2}}$, with $a,b \in \mathbb{R}$, $a>b>0$

A problem from XIV GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN The correspondence round, 2018

Diophantine Equation Concerning Powers of 2

Interesting 3 Variable Inequality for Real Numbers

Math Olympiad exercise $\frac{x}{\sqrt{3x+2y+z}}+\frac{y}{\sqrt{3y+2z+x}}+\frac{z}{\sqrt{3z+2x+y}}\leq\sqrt{\frac{x+y+z}{2}}$

Finding unit digit of $f(10)$

Symmetric inequality with three variables including radicals

Number of ways to gain $10$ points while gaining conditional $1$, $2$, or $3$ points at a time?

Number of $6$-digit numbers made up of the digits $1$, $2$, and $3$ with no digit occurring $3$ or more times consecutively?

Proving an Inequality - Functions

Finding the minimum of $λ$ such that $|a-b|^p\le λ(2-|a-b|)\left|\frac{a|a|^{p-1}}{1-|a|}-\frac{b|b|^{p-1}}{1-|b|}\right|$ for $a,b\in(-1,1)$

Prove: $(a+b) \sqrt {ab}+(a+c) \sqrt {ac}+(b+c)\sqrt {bc}≥ \frac {(a+b+c)^2}{2}$

Writing integers as a product of as few elements of $\{\frac21, \frac32, \frac43, \frac54, \ldots\}$ as possible

Converse of theorem of two tangent circles

Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $ f(x) \geq f(y)\sin(x-y)$

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