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Problem 1412

AIME late

Combinatorics Difficulty 5.8 Multiple choice Italian Mathematical Olympiad · Italy

A ski competition is divided into two runs; an athlete placed $3^{\circ}$ in the first and $5^{\circ}$ in the second. Knowing that the final ranking is drawn up on the basis of the sum of the times obtained in the individual runs, that there are 70 competitors and assuming that there were no ties, state which positions the athlete can occupy in the final ranking.

Pick one

Official solution

The answer is (D). Indeed, since only 2 did a better time in the first run and only four did better than him in the second run, the athlete in question obtained in both runs times better than the remaining 63 athletes, so he could not have ranked higher than seventh. To verify that all positions, from first to seventh place, are attainable, let us denote by $A, B, C, D, E$ and $F$ six skiers' names and by $X$ our athlete. An example of how $X$ can end up seventh is given by these two runs:

	$A$	$B$	$C$	$D$	$E$	$F$	$X$
$1^{o}$ manche	$1:29$	$1:30$	$1:59$	$1:58$	$1:57$	$1:56$	$1:55$
$2^{o}$ manche	$1:58$	$1:59$	$1:27$	$1:26$	$1:25$	$1:24$	$1:57$
totale	$3:27$	$3:29$	$3:36$	$3:34$	$3:32$	$3:30$	$3:52$

If instead

X

had done

1:31

in the first and

1:28

in the second run, with a total of

2:59

he would have come first, while still remaining third and fifth in the two runs. One can then check without too much difficulty that, leaving unchanged the times of

A, B, C, D, E, F

, and modifying only those of

X

, the latter can end up in any position between first and seventh.

Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty, ordering) added by this project.