Questions tagged [buffalo-way]

• For example:

Prove that $$\sum_{cyc}(x^3-x^2y-x^2z+xyz)\geq0$$ for non-negatives $x$, $y$ and $z$.

A proof by BW:

Let $x$ be a minimal number between numbers $x$, $y$ and $z$, $y=x+u$ and $z=x+v$.

Hence, $$\sum_{cyc}(x^3-x^2y-x^2z+xyz)=(u^2-uv+v^2)x+(u+v)(u-v)^2\geq0.$$

Sometimes it is better to use $y=x+u$, $z=x+u+v$ or $y=x+u+v$, $z=x+u$,

where $u$ and $v$ are non-negatives.

• If $x$, $y$ and $z$ are sides-lengths of triangle and $z=\max\{x,y,z\}$, we can use the following substitution:

$x=a+u$, $y=a+v$ and $z=a+u+v$, where $a>0$, $u\geq0$ and $v\geq0$.

For example:

For all triangle prove that: $$a^3b^2+b^3c^2+c^3a^2\geq(a^2+b^2+c^2)abc$$

Proof:

Let $c$ be a maximal number between numbers $a$, $b$ and $c$, $a=x+u$, $b=x+v$ and $c=x+u+v$, where $x>0$, $u$ and $v$ are non-negatives.

Hence, $$a^3b^2+b^3c^2+c^3a^2-(a^2+b^2+c^2)abc=(u^2-uv+v^2)x^3+$$ $$+3(u^3-uv^2+v^3)x^2+(3u^4+2u^3v-3u^2v^2-2uv^3+3v^4)x+u^5+u^4v-2u^2v^3+v^5\geq0.$$

• Another example.

For non-negatives $a$, $b$, $c$ and $d$ prove that: $$a^4+b^4+c^4+d^4+4abcd\geq2(a^2bc+b^2cd+c^2da+d^2ab).$$

A proof:

Let $a=\min\{a,b,c,d\}$, $b=a+u$, $c=a+v$ and $d=a+w$.

Hence, $$a^4+b^4+c^4+d^4+4abcd-2(a^2bc+b^2cd+c^2da+d^2ab)=$$ $$=2(2u^2+2v^2+2w^2-uv-uw-vw)a^2+$$ $$+2(2u^3+2v^3+2w^3-u^2v-u^2w-v^2w-w^2u)a+u^4+v^4+w^4-2u^2vw\geq0$$