## Search found 64 matches

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

### HKTST 2016

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

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