Question

Para cualesquiera tres números reales $a,$ $b,$ y $c,$ con $b \neq c,$ la operación $\diamond$ se define por $\diamond(a, b, c) = \dfrac{a}{b - c}.$ ¿Cuál es el valor de $\diamond(\diamond(1, 2, 3), \diamond(2, 3, 1), \diamond(3, 1, 2))$ ?

For any three real numbers $a,$ $b,$ and $c,$ with $b \neq c,$ the operation $\diamond$ is defined by $\diamond(a, b, c) = \dfrac{a}{b - c}.$ What is $\diamond(\diamond(1, 2, 3), \diamond(2, 3, 1), \diamond(3, 1, 2))?$

$-\dfrac{1}{2}$

$-\dfrac{1}{4}$

$0$

$\dfrac{1}{4}$

$\dfrac{1}{2}$

Solución:

Los valores internos son $\begin{aligned} \diamond(1,2,3) &= \dfrac{1}{2-3} = -1, \\ \diamond(2,3,1) &= \dfrac{2}{3-1} = 1, \\ \diamond(3,1,2) &= \dfrac{3}{1-2} = -3. \end{aligned}$

Por lo tanto $\begin{aligned} \diamond(-1, 1, -3) &= \dfrac{-1}{1 - (-3)} \\ &= -\dfrac{1}{4}. \end{aligned}$

Por lo tanto, la respuesta correcta es B.

The inner values are $\begin{aligned} \diamond(1,2,3) &= \dfrac{1}{2-3} = -1, \\ \diamond(2,3,1) &= \dfrac{2}{3-1} = 1, \\ \diamond(3,1,2) &= \dfrac{3}{1-2} = -3. \end{aligned}$

Therefore $\begin{aligned} \diamond(-1, 1, -3) &= \dfrac{-1}{1 - (-3)} \\ &= -\dfrac{1}{4}. \end{aligned}$

Thus, the correct answer is B.

2004 AMC 10A Problema 2

Solución:

El Problema 2 en otros años