solution: Note that $p(x)$ has degree at most 2.  Also, $p(a) = p(b) = p(c) = 1.$  Thus, the polynomials $p(x)$ and 1 agree at three different values, so by the Identity Theorem, they are the same polynomial.  Hence, the degree of $p(x)$ (which is the constant polynomial 1) is $\boxed{0}.$
  
  You can check manually that
  \[p(x) = \frac{(x - b)(x - c)}{(a - b)(a - c)} + \frac{(x - a)(x - c)}{(b - a)(b - c)} + \frac{(x - a)(x - b)}{(c - a)(c - b)}\]simplifies to 1.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Let $a,$ $b,$ and $c$ be distinct real numbers.  Find the degree of the polynomial
  \[p(x) = \frac{(x - b)(x - c)}{(a - b)(a - c)} + \frac{(x - a)(x - c)}{(b - a)(b - c)} + \frac{(x - a)(x - b)}{(c - a)(c - b)}.\]
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 0
extra_info.question: Let $a,$ $b,$ and $c$ be distinct real numbers.  Find the degree of the polynomial
  \[p(x) = \frac{(x - b)(x - c)}{(a - b)(a - c)} + \frac{(x - a)(x - c)}{(b - a)(b - c)} + \frac{(x - a)(x - b)}{(c - a)(c - b)}.\]
extra_info.split: test