Let $a$, $b$, $c$, and $n$ be positive integers. If $a + b + c = 19 \cdot 97$ and \[a + n = b - n = \frac{c}{n},\] compute the value of $a$.
+128450 a + n = c/n b - n = c/n
a = c/n - n b = c/n + n
So
a + b + c = 19 * 97
c/n - n + c/n + n + c = 19 * 97
2c/n + c = 1843
2c + cn = 1843n
c ( 2 + n) = 1843n
c = 1843n / ( 2 + n)
When n = 17, c = 1649
a = c/n - n = 1649/17 - 17 = 97 - 17 = 80
b= c/n + n = 1649/17 + 17 = 97 + 17 = 114
a + b + c = 80 + 114 + 1649 = 97 * 19 = 1843
