# Questions tagged [buffalo-way]

Inequalities that can be proved by BW (Buffalo Way).

142 questions
Filter by
Sorted by
Tagged with
80 views

194 views

1 vote
145 views

• 3,597
1 vote
62 views

1 vote
87 views

• 409
166 views

### How to prove $\frac{ab^2}{1+2b^2+c^2}+\frac{bc^2}{1+2c^2+a^2}+\frac{ca^2}{1+2a^2+b^2} \le \frac{3}{4}$ if $a+b+c=3$

$a,b,c\ge 0,a+b+c=3.$ Prove: $$\frac{ab^2}{1+2b^2+c^2}+\frac{bc^2}{1+2c^2+a^2}+\frac{ca^2}{1+2a^2+b^2} \le \frac{3}{4}$$ This problem was found in this post. As you can see, no one in that post gave ...
1 vote
77 views

### For any positive real numbers $a,b,c$, show that $\min\{(b-c)^2,(c-a)^2,(a-b)^2)\} \leq \frac{a^2+b^2+c^2}{5}$

For any positive real numbers $a,b,c$, show that $\min\{(b-c)^2,(c-a)^2,(a-b)^2)\} \leq \frac{a^2+b^2+c^2}{5}$ I managed to show this with a 3 instead of a 5 : The inequality is symmetric in $a$, $b$,...
• 11
97 views

95 views

### Prove $\sum\limits_{cyc} \frac{(a+14)^2}{(b+4)^2}\geq \frac{3}{2} \sum\limits_{cyc} \left(\frac{a+14}{b+4}+\frac{b+4}{a+14}\right)-6$

Problem. For $a,b,c\geq 0.$ Prove that: $$\sum\limits_{cyc} \frac{(a+14)^2}{(b+4)^2}\geq \frac{3}{2} \sum\limits_{cyc} \left(\frac{a+14}{b+4}+\frac{b+4}{a+14}\right)-6$$ I have a solution but it's not ...
• 2,001
344 views

### Prove $\sqrt{a + ab} + \sqrt{b} + \sqrt{c} \ge 3$ for $c = \min(a, b, c)$ and $ab + bc + ca = 2$

Problem 1: Let $a, b, c \ge 0$ with $c = \min(a, b, c)$ and $ab + bc + ca = 2$. Prove that $\sqrt{a + ab} + \sqrt{b} + \sqrt{c} \ge 3$. Background: This problem was proposed by csav10@AoPS (https://...
• 39k
99 views

### Prove $\sum_{\mathrm{cyc}} \sqrt{34x^2 + 28y^2 + 7z^2 - xy - 28yz + 41zx} \ge 9x + 9y + 9z$

Problem 1: Let $x, y, z \ge 0$. Prove that $$\sum_{\mathrm{cyc}} \sqrt{34x^2 + 28y^2 + 7z^2 - xy - 28yz + 41zx} \ge 9x + 9y + 9z. \tag{1}$$ Background: I came up with the problem when I tried to ...
• 39k
144 views

### Prove $5\Big(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\Big)\geq \frac{a^2+b^2+c^2}{ab+bc+ca}+10.$

Problem. (?) For $a,b,c$ be non-negative numbers such as $a \geq 2(b+c).$ Prove:$$5\Big(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\Big)\geq \frac{a^2+b^2+c^2}{ab+bc+ca}+10.$$ My Solution. We write the ...
• 2,001
800 views

• 2,001
196 views

### Proving ${\frac {35{x}^{2}+7x(y+z)+23yz}{35(x^2+y^2+z^2)+37(xy+yz+zx)}}\leqslant \sqrt {{\frac {{x}^{2}+yz}{6\,{y}^{2} +6\,yz+6\,{z}^{2}}}}$

For $x,y,z \geqslant 0.$ Proving$:$ $${\dfrac {35{x}^{2}+7x(y+z)+23yz}{35(x^2+y^2+z^2)+37(xy+yz+zx)}}\leqslant \sqrt {{\dfrac {{x}^{2}+yz}{6{y}^{2} +6yz+6{z}^{2}}}}\quad \quad(\text{tthnew})$$ My ...
• 2,001
143 views

304 views

### Proving $3(a^4 + b^4 + c^4 + d^4) + 4abcd \geq (a+b+c+d)(a^3 + b^3 + c^3 + d^3)$

Let $a,b,c,d>0$. Prove that$:$ $$3(a^4 + b^4 + c^4 + d^4) + 4abcd \geqslant (\,a+b+c+d\,)(\,a^3 + b^3 + c^3 + d^{3}\,)$$ As pointed out by @tthnew, this question was posted in: https://...
100 views

### Proving $\displaystyle \sum_{cyc}\frac{(a^2+b^2)}{a+b}\leqslant \frac{3(a^2+b^2+c^2)}{a+b+c}$ [duplicate]

For $a,b,c>0$, prove $\displaystyle \sum_{cyc}\frac{(a^2+b^2)}{a+b}\leqslant \frac{3(a^2+b^2+c^2)}{a+b+c}$ I've simplified the inequality by multiplying both sides with $(a+b+c).$ So the inequality ...
• 521
175 views

### Proving $\frac {a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a} \geqslant \frac 32 \cdot \sqrt[6]{\frac{ab+bc+ca}{a^2+b^2+c^2}}$

For $a,b,c>0.$ Prove$:$ $$\displaystyle \frac {a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a} \geqslant \frac 32 \cdot \sqrt[6]{\dfrac{ab+bc+ca}{a^2+b^2+c^2}}$$ My try. The Buffalo Way method help here$,$ but ...
• 2,001
80 views

### on possible generalisations of $1\le\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\le 2$

Here is a known double inequality for positive numbers: $$1\le\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\le 2$$ Source: https://www.cut-the-knot.org/m/Algebra/Crux4196.shtml I'm curious if at least ...
• 156
1 vote