Baltic Way 1996 · Problem 11

Algebra

The real numbers $x_{1}, x_{2}, \ldots, x_{1996}$ have the following property: for any polynomial $W$ of degree 2 at least three of the numbers $W\left(x_{1}\right), W\left(x_{2}\right), \ldots, W\left(x_{1996}\right)$ are equal. Prove that at least three of the numbers $x_{1}, x_{2}, \ldots, x_{1996}$ are equal.

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Polynomials

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Solution

Solution:

Let $m = \min \{ x_{1}, \ldots, x_{1996} \}$ . Then the polynomial $W(x) = (x - m)^{2}$ is strictly increasing for $x \geq m$ . Hence if $W\left(x_{i}\right) = W\left(x_{j}\right)$ we must have $x_{i} = x_{j}$ , and the conclusion follows.

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Results from Baltic Way 1996

10 teams

Mean score: 4.5 / 5
Scores of 4 or 5: 9 / 10
Estonia: 5 / 5

Score distribution

All team scores

Team	Score
Poland	5 / 5
Latvia	5 / 5
Sweden	5 / 5
Denmark	5 / 5
St. Petersburg	5 / 5
Finland	5 / 5
Norway	5 / 5
Lithuania	5 / 5
Estonia	5 / 5
Iceland	0 / 5

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