By adding the inequalities term by term \( x + y + z \leq t \), \( -2x^2 - 2y^2 - 2z^2 \leq -2t \), and \( x^3 + y^3 + z^3 \leq t \), we obtain: \[ x(1-x)^2 + y(1-y)^2 + z(1-z)^2 \leq 0 \] Since \( x, y, z \in [0, \infty) \), it follows that \( x, y, z \in \{0, 1\} \). - If \( x = y = z = 0 \), then \( t = 0 \). - If exactly two of the numbers \( x, y, z \) are zero, then \( 1 \leq t \geq 1 \), hence \( t = 1 \) and \( (x, y, z) \in \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\} \). - If exactly one of the numbers \( x, y, z \) is zero, then \( 2 \leq t \geq 2 \), so \( t = 2 \) and \( (x, y, z) \in \{(1, 1, 0), (1, 0, 1), (0, 1, 1)\} \). - If \( x = y = z = 1 \), we deduce that \( t = 3 \).