2001 IMO Shortlist Problems/N6

Problem

Is it possible to find 100 positive integers not exceeding 25,000, such that all pairwise sums of them are different?

Solution

For example, let's say that the integers are, $k, k+a_1, k+a_2, ..., k+a_{99}$ . Now this turns into a problem of solving for the $99$ integers $a_i$ . This then each ai takes on the form, $j+b_1, j+b_2,..., j+b_{98}$ . Then we must find the $98$ $b$ integers. By doing this process over and over again, we obtain the last $3$ numbers, $y, y+u_1, y+u_2$ . Obviously these 3 integers can have different sums, and the number of different "parts" in every sequence (the number of terms that are different for $a_i, b_i, c_i$ , etc.) is $99+98+\ldots+2$ , not exceeding $25000$ .

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2001 IMO Shortlist Problems/N6

Problem

Solution

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