Course overview

Resolver el sistema

$$\frac{b(x+y)}{x+y+cxy} + \frac{c(z+x)}{x+z+bxz} = a$$

$$\frac{c(y+z)}{y+z+ayz} + \frac{a(x+y)}{x+y+cxy} = b$$

$$\frac{a(z+x)}{z+x+bxz} + \frac{b(y+z)}{y+z+ayz} = c$$

Eliminar $a$, $b$, $c$ del sistema

$$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$

$$a^2+b^2+c^2=1$$

$$a+b+c=1.$$

Dado

$$\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \alpha$$

$$\frac{x}{z} + \frac{y}{x} + \frac{z}{y} = \beta$$

$$\left(\frac{x}{y} + \frac{y}{z}\right) \left(\frac{y}{z} + \frac{z}{x}\right) \left(\frac{z}{x} + \frac{x}{y}\right) = \gamma.$$

Eliminar $x$, $y$ y $z$.

Demostrar que si

$$x+y+z+w=0$$

$$ax+by+cz+dw=0$$

$$(a-d)^2(b-c)^2(xw+yz) + (b-d)^2(c-a)^2(yw+zx) + (c-d)^2(a-b)^2(zw+xy) = 0,$$

entonces

$$\frac{x}{(d-b)(d-c)(b-c)} = \frac{y}{(d-c)(d-a)(c-a)} = \frac{z}{(d-a)(d-b)(a-b)} = \frac{w}{(b-c)(c-a)(a-b)}.$$

Encontrar los valores positivos de las incógnitas $x$, $y$, $u$ y $v$ que satisfacen el sistema

$$u^p v^q = a^x, \quad u^q v^p = a^y, \quad u^2 v^y = b, \quad u^y v^x = c$$

$$(a, b, c > 0 \text{ y } p^2 - q^2 \neq 0).$$