Let $a,b,c$ be positive real numbers satisfying $abc=1$. Determine the smallest possible value of

$$\frac{a^3+8}{a^3(b+c)}+\frac{b^3+8}{b^3(a+c)}+\frac{c^3+8}{c^3(b+a)}$$

## Search found 64 matches

- Thu Mar 23, 2017 11:49 pm
- Forum: Algebra
- Topic: HKTST 2016
- Replies:
**1** - Views:
**3971**

- Thu Mar 23, 2017 11:36 pm
- Forum: Algebra
- Topic: $2009$ USA TST Inequality
- Replies:
**1** - Views:
**1357**

### Re: $2009$ USA TST Inequality

Just substitute $x = 1/a$, $y = 1/b$ and $z = 1/c$.

The rest is so cool.

The rest is so cool.

- Mon Mar 20, 2017 10:30 pm
- Forum: Number Theory
- Topic: Equation
- Replies:
**1** - Views:
**1327**

### Re: Equation

x^2+xy+y^2=(x+y+3)^3/27,find all(x,y) It is obvious that $x+y$ is divisible by $3$ . Put $x+y$ = $3k$ . So, the equation becomes, $$(3k)^2 - x(3k-x) = (k+1)^3 $$ => $$ x^2 - 3kx + 9k^2 = k^3 + 3k^2 + 3k +1 $$ => $$ x^2 - 3kx - k^3 + 6k^2 - 3k -1 = 0 $$ As we have to find the integer roots of this e...

- Fri Aug 19, 2016 8:17 pm
- Forum: Geometry
- Topic: APMO 2013 P5
- Replies:
**3** - Views:
**2225**

### APMO 2013 P5

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Pr...