## Task B-1.6.

Three brothers were born one after another with the same age difference. When the youngest brother was born, the oldest was six years younger than the middle brother is today. The difference between the squares of the ages of the oldest and the youngest brother is 432. How old is each of the three brothers today?
## Solution.

Let the age of the middle brother be denoted by $x$, and the age difference by $a$. Then the youngest brother is $x-a$ years old, and the oldest is $x+a$.

1 point

When the youngest was born, the oldest was $x+a-(x-a)=2a$ years old, which is 6 years less than $x$.

$2a = x - 6 \Rightarrow x = 2a + 6$ 1 point

$(x+a)^2 - (x-a)^2 = 432$

1 point

$ax = 108$

1 point

$a(2a + 6) = 108$

$a^2 + 3a - 54 = 0$

1 point

$a^2 + 9a - 6a - 54 = 0$

$a(a + 9) - 6(a + 9) = 0$

$(a + 9)(a - 6) = 0$ 1 point

$a = 6$

1 point

The age difference is 6.

$x = 2a + 6 = 18$

The youngest brother is 12 years old, the middle brother is 18 years old, and the oldest is 24 years old.