Question

Cuando se añade $15$ a una lista de enteros, la media aumenta en $2.$ Cuando se añade $1$ a la lista ampliada, la media de la lista ampliada disminuye en $1.$ ¿Cuántos enteros había en la lista original?

When $15$ is appended to a list of integers, the mean is increased by $2.$ When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1.$ How many integers were in the original list?

$4$

$5$

$6$

$7$

$8$

Solución:

Sea la lista original con $n$ enteros y media $m,$ así que su suma es $mn.$ Añadir $15$ da $\begin{aligned} &(m + 2)(n + 1) \\ &= mn + 15 \\ &\implies m + 2n = 13. \end{aligned}$

Añadir $1$ a esa lista ampliada da $\begin{aligned} &(m + 1)(n + 2) \\ &= mn + 16 \\ &\implies 2m + n = 14. \end{aligned}$

Resolviendo $m + 2n = 13$ y $2m + n = 14$ se obtiene $m = 5$ y $n = 4.$

Por lo tanto, la respuesta correcta es A.

Let the original list have $n$ integers with mean $m,$ so its sum is $mn.$ Appending $15$ gives $\begin{aligned} &(m + 2)(n + 1) \\ &= mn + 15 \\ &\implies m + 2n = 13. \end{aligned}$

Appending $1$ to that enlarged list gives $\begin{aligned} &(m + 1)(n + 2) \\ &= mn + 16 \\ &\implies 2m + n = 14. \end{aligned}$

Solving $m + 2n = 13$ and $2m + n = 14$ yields $m = 5$ and $n = 4.$

Thus, the correct answer is A.

2002 AMC 10B Problema 25

Solución:

El Problema 25 en otros años