Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.9 AIME, harder Prove it Russia

A collection of $n$ real numbers is written on the blackboard. It appears that the square of each written number is greater than the product of any two other written numbers. Find the greatest possible value of $n$ . (I. Bogdanov)

На доске написаны несколько чисел. Известно, что квадрат любого записанного числа больше произведения любых двух других записанных чисел. Какое наибольшее количество чисел может быть на доске?
(И. Богданов)

Solution

$n = 3$ .

Suppose there are at least four numbers, and let $a$ be the number with the smallest absolute value. Among the remaining numbers, at least two have the same sign (both nonnegative or both nonpositive). Denote them by $b$ and $c$ ; then $bc = |bc| \geq |a|^2 = a^2$ , which contradicts the condition.

It remains to give an example of three numbers satisfying the condition. For example, the numbers $1$ , $2$ , $-3$ are suitable.

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