Question

How do I show that $\sum_{\text{cyc}}\frac{3a^2-2ab+3b^2}{(a+b)^2} < \frac{9}{4}(\frac{a}{c} + \frac{c}{a}) - \frac{3}{2}$?

score 153 · Answer 1 · 2026-04-14 17:42:08

153

Views

How do I show that $\sum_{\text{cyc}}\frac{3a^2-2ab+3b^2}{(a+b)^2} < \frac{9}{4}(\frac{a}{c} + \frac{c}{a}) - \frac{3}{2}$?

Published on 14 Apr 2026 - 17:42

score 259 · Answer 2 · 2026-04-12 14:10:29

259

Views

Doubt in IMO $1993$ Problem 1

Published on 12 Apr 2026 - 14:10

score 46 · Answer 3 · 2026-04-13 22:15:14

46

Views

The probability that $n$ tosses produce no sequence of $HHH$

Published on 13 Apr 2026 - 22:15

score 9.5k · Answer 4 · 2026-04-16 01:17:43

9.5k

Views

Prove $\sqrt{ xy} \leq \frac{x + y}{2}$ for all positive $x$ and $y$

Published on 16 Apr 2026 - 1:17

score 198 · Answer 5 · 2026-04-15 19:07:15

198

Views

Proving $f^{-1}(G\cup H)=f^{-1}(G) \cup f(H)$ and $f^{-1}(G\cap H)= f^{-1}(G) \cap f^{-1}(H)$

Published on 15 Apr 2026 - 19:07

score 40 · Answer 6 · 2026-04-16 05:25:28

40

Views

If f is injective then $f•f^{-1}(y)=y$ for all $y\in R(f)$

Published on 16 Apr 2026 - 5:25

score 51 · Answer 7 · 2026-04-13 03:44:00

51

Views

How can this function be the connecting homomorphism of the Mayer-Vietoris sequence if it's not composed of the same functions?

Published on 13 Apr 2026 - 3:44

score 122 · Answer 8 · 2026-04-16 16:10:25

122

Views

Proof explanation: $|X|=|Y|=m \in \mathbb{N}$ there exists exactly $m! = 1 \cdot 2 \cdot \ldots \cdot m$

Published on 16 Apr 2026 - 16:10

score 676 · Answer 9 · 2026-04-16 04:38:20

676

Views

The polynomial $x^{2n} - 2x^{2n-1} + 3x^{2n-2}- \cdots -2nx + 2n + 1$ has no real roots.

Published on 16 Apr 2026 - 4:38

score 65 · Answer 10 · 2026-04-08 21:06:15

65

Views

Doubt from Maximum principle Elliptic operator

Published on 08 Apr 2026 - 21:06

score 47 · Answer 11 · 2026-03-25 19:10:12

47

Views

In a random graph $G(n,p)$ with $\delta(G) \geq \delta$, how is the event $\{S \subset V \text{ is not a dominating set} \}$ characterized?

Published on 25 Mar 2026 - 19:10

score 34 · Answer 12 · 2026-04-14 23:04:49

34

Views

Help understanding how to prove that $S = \{ \frac{1}{n+1} : n\in \mathbb{N}\}$ has $Sup(S) = \frac{1}{2}$

Published on 14 Apr 2026 - 23:04

score 40 · Answer 13 · 2026-04-10 15:00:25

40

Views

Counting arguments for infinitely many primes

Published on 10 Apr 2026 - 15:00

score 85 · Answer 14 · 2026-04-10 07:09:50

85

Views

Explanation of Proof of Theorem 43 of Chapter 4 of Protter's Stochastic Integration

Published on 10 Apr 2026 - 7:09

score 96 · Answer 15 · 2026-04-15 16:02:46

96

Views

Prove if $|x|=mn$ and gcd$(a,m)=1$ then $m$ divides $|x^a|$

Published on 15 Apr 2026 - 16:02

List Question

How do I show that $\sum_{\text{cyc}}\frac{3a^2-2ab+3b^2}{(a+b)^2} < \frac{9}{4}(\frac{a}{c} + \frac{c}{a}) - \frac{3}{2}$?

Doubt in IMO $1993$ Problem 1

The probability that $n$ tosses produce no sequence of $HHH$

Prove $\sqrt{ xy} \leq \frac{x + y}{2}$ for all positive $x$ and $y$

Proving $f^{-1}(G\cup H)=f^{-1}(G) \cup f(H)$ and $f^{-1}(G\cap H)= f^{-1}(G) \cap f^{-1}(H)$

If f is injective then $f•f^{-1}(y)=y$ for all $y\in R(f)$

How can this function be the connecting homomorphism of the Mayer-Vietoris sequence if it's not composed of the same functions?

Proof explanation: $|X|=|Y|=m \in \mathbb{N}$ there exists exactly $m! = 1 \cdot 2 \cdot \ldots \cdot m$

The polynomial $x^{2n} - 2x^{2n-1} + 3x^{2n-2}- \cdots -2nx + 2n + 1$ has no real roots.

Doubt from Maximum principle Elliptic operator

In a random graph $G(n,p)$ with $\delta(G) \geq \delta$, how is the event $\{S \subset V \text{ is not a dominating set} \}$ characterized?

Help understanding how to prove that $S = \{ \frac{1}{n+1} : n\in \mathbb{N}\}$ has $Sup(S) = \frac{1}{2}$

Counting arguments for infinitely many primes

Explanation of Proof of Theorem 43 of Chapter 4 of Protter's Stochastic Integration

Prove if $|x|=mn$ and gcd$(a,m)=1$ then $m$ divides $|x^a|$

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