Question

IMO 1984: Prove that $0 ≤ yz + zx +xy −2xyz ≤ \frac {7}{27}$, where $x,y$ and $z$ are non-negative real numbers for which $x + y + z = 1.$

score 2.3k · Answer 1 · 2019-05-30 07:40:38

2.3k

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IMO 1984: Prove that $0 ≤ yz + zx +xy −2xyz ≤ \frac {7}{27}$, where $x,y$ and $z$ are non-negative real numbers for which $x + y + z = 1.$

Published on 30 May 2019 - 7:40

score 145 · Answer 2 · 2019-05-30 08:15:13

145

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Among any $5$ vertices there is a vertex connected to $4$ others. What is the minimum number of edges in such a graph with 40 vertices?

Published on 30 May 2019 - 8:15

score 120 · Answer 3 · 2019-05-30 14:27:20

120

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Is this a valid solution to Putnam 2017 A3?

Published on 30 May 2019 - 14:27

score 77 · Answer 4 · 2026-03-26 03:01:28

77

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Solve $x+y+z=3$; $x^3+y^3+z^3=15$; $x^5+y^5+z^5=83$

Published on 26 Mar 2026 - 3:01

score 118 · Answer 5 · 2019-05-31 12:29:54

118

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Olympiad inequality $\Big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\Big)\sqrt{\frac{\prod_{cyc}(49x-7y+z)}{43^3}}\leq \sqrt{3(x+y+z)}$

Published on 31 May 2019 - 12:29

score 151 · Answer 6 · 2019-05-31 16:28:34

151

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An olympiad question

Published on 31 May 2019 - 16:28

score 241 · Answer 7 · 2019-06-02 16:37:09

241

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30 languages spoken, how many participants are at the conference?

Published on 02 Jun 2019 - 16:37

score 204 · Answer 8 · 2026-03-26 02:55:02

204

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Rationalizing $\frac{1}{\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}}$

Published on 26 Mar 2026 - 2:55

score 457 · Answer 9 · 2019-06-04 04:15:34

457

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Advanced algebra-precalculus

Published on 04 Jun 2019 - 4:15

score 65 · Answer 10 · 2019-06-04 18:49:35

65

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SMO 2012 Question 7

Published on 04 Jun 2019 - 18:49

score 464 · Answer 11 · 2026-03-29 00:50:01

464

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For $a,b,c,d > 0$ and $abcd = 1$, show that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{12}{a + b + c + d} \geq 7$

Published on 29 Mar 2026 - 0:50

score 104 · Answer 12 · 2026-03-28 15:55:56

104

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Prove that $n$ is Divisible by $6$ when the Quadratics have integer roots

Published on 28 Mar 2026 - 15:55

score 64 · Answer 13 · 2019-06-06 22:52:11

64

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Getting into competition math for an eager to learn high school student

Published on 06 Jun 2019 - 22:52

score 118 · Answer 14 · 2026-03-25 06:10:55

118

Views

Nice refinement of an inequality by Michael Rozenberg

Published on 25 Mar 2026 - 6:10

score 50 · Answer 15 · 2019-06-07 16:59:09

50

Views

Suppose that * is an associative operation on a set S.

Published on 07 Jun 2019 - 16:59

List Question

IMO 1984: Prove that $0 ≤ yz + zx +xy −2xyz ≤ \frac {7}{27}$, where $x,y$ and $z$ are non-negative real numbers for which $x + y + z = 1.$

Among any $5$ vertices there is a vertex connected to $4$ others. What is the minimum number of edges in such a graph with 40 vertices?

Is this a valid solution to Putnam 2017 A3?

Solve $x+y+z=3$; $x^3+y^3+z^3=15$; $x^5+y^5+z^5=83$

Olympiad inequality $\Big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\Big)\sqrt{\frac{\prod_{cyc}(49x-7y+z)}{43^3}}\leq \sqrt{3(x+y+z)}$

An olympiad question

30 languages spoken, how many participants are at the conference?

Rationalizing $\frac{1}{\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}}$

Advanced algebra-precalculus

SMO 2012 Question 7

For $a,b,c,d > 0$ and $abcd = 1$, show that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{12}{a + b + c + d} \geq 7$

Prove that $n$ is Divisible by $6$ when the Quadratics have integer roots

Getting into competition math for an eager to learn high school student

Nice refinement of an inequality by Michael Rozenberg

Suppose that * is an associative operation on a set S.

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