Question

For $t = 1, 2, 3, 4,$ define $S_t = \sum_{i=1}^{350} a_i^t,$ where $a_i \in \{1, 2, 3, 4\}.$ If $S_1 = 513$ and $S_4 = 4745,$ find the minimum possible value for $S_2.$

Answer

Solution:

For $j = 1, 2, 3, 4,$ let $m_j$ be the number of $a_i$ equal to $j.$ Then $m_1 + m_2 + m_3 + m_4 = 350, \quad m_1 + 2m_2 + 3m_3 + 4m_4 = 513, \quad m_1 + 16m_2 + 81m_3 + 256m_4 = 4745.$

Subtracting the first equation from the other two gives $m_2 + 2m_3 + 3m_4 = 163$ and $15m_2 + 80m_3 + 255m_4 = 4395;$ subtracting $15$ times the former from the latter leaves $50m_3 + 210m_4 = 1950,$ that is, $5m_3 + 21m_4 = 195.$ Hence $m_4$ is a nonnegative multiple of $5,$ and only $m_4 = 0$ and $m_4 = 5$ keep everything nonnegative, giving $(m_1, m_2, m_3, m_4) = (226, 85, 39, 0)$ or $(215, 112, 18, 5).$

These yield $S_2 = m_1 + 4m_2 + 9m_3 + 16m_4 = 917$ and $905$ respectively, so the minimum is $905.$

2009 AIME I Problem 14

Solution:

Problem 14 in Other Years