Problem #PRU-79473

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebra and arithmetic Algebraic inequalities and systems of inequalities Quadratic inequalites (several variables)

13–14 3.0

Problem

The numbers $a_{1}, a_{2}, \dots, a_{1985}$ are the numbers $1, 2, \dots, 1985$ rearranged in some order. Each number $a_{k}$ is multiplied by its number $k$ , and then the largest number is chosen among the resulting 1985 products. Prove that it is not less than $993^{2}$ .

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