Let a1, a2, ... , a52 be positive integers such that a1 < a2 < ... < a

Let \(a_1\), \(a_2\), ... , \(a_{52}\) be positive integers such that \(a_1\)<\(a_2\)<...<\(a_{52}\). Suppose, their arithmetic mean is one less than the arithmetic mean of \(a_2\), \(a_3\), ..., \(a_{52}\). If \(a_{52}=100\), then the largest possible value of \(a_1\) is

\(\frac{(a_1 + a_2 +...+ a_{52})}{52} +1 = \frac{(a_2 +a _3+...+ a_{52})}{51 }\)
\(\frac{(a_1 + a_2 +...+ a_{52}+ 52)}{52}= \frac{(a_2 +a _3+...+ a_{52})}{51}\)
\(51 (a_1 + a_2 +...+ a_{52}+ 52) =52 (a_2 +a _3+...+ a_{52})\)
--> \(51a _1 + 51*52= a_2 +a _3+...+ a_{52}\)
\(a_{52} =100\)

In order \(a _1\) to be the largest possible value,
\(a_2 +a _3+...+ a_{52} = 50 +51 +52 +...+100 = \frac{(100+50)*51}{2}= 75*51\)
--> \(51a _1 + 51*52= 75*51\)
\(a _1= 75 -52 = 23\)

Answer (B).