The sum of x distinct integers greater than zero is less than 75. What is the-greatest possible value of x?

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Solution

Integers are positive and distinct.
$\frac{x (x + 1)}{2} < 75 x^{2} + x - 150 < 0 = x = \frac{- 1 \pm \sqrt{1 + 600}}{2} = \frac{\sqrt{601} - 1}{2} x \in [0, \frac{\sqrt{601} - 1}{2}] (x > 0)$
Greatest possible value of $x = 11$

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