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121 women are competing in the heptathlon in Olympic Games.

If British track and field star Jessica Ennis finishes in 104th place in every event, what is the highest place she can finish in the overall heptathlon?

PS: Feel free to google around for the information that you think might be essential to solving the problem.

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    $\begingroup$ Could you describe how exactly the rating system in olympic heptathlon works? $\endgroup$
    – Philipp
    Commented Mar 2, 2015 at 13:02
  • $\begingroup$ @Philipp See here for point system. $\endgroup$
    – Donbhupi
    Commented Mar 2, 2015 at 13:07

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The best she can hope for is 2nd place.

Let's assume that in each event, there is a 103-way tie for first place. Jessica finishes 104th with a score fractionally below the leaders, and the remaining 7 competitors score precisely zero points.

Let's also assume that the winning scores in each event are equal to $X_i$ for $(1 \le i \le 7)$. So competitors who finish among the joint winners in all seven events will achieve a total score of $T = \sum^7_{i=1} X_i$, and Jessica's total score will be $(T - \epsilon)$ points, where $\epsilon$ is some small value.

If $m$ is the event where the joint winners achieved the highest score (i.e., $X_m > X_i \ \ \forall i \not= m)$, then Jessica will achieve the best position if as many as possible of the winners in event $m$ score zero in at least one other event. Since there are seventeen possible places that they can take in the remaining six events, that means Jessica can beat at most 102 of them.

So her best possible finishing position is $104 - 102 = 2$.

(Thanks to Duncan for pointing out the glaring error in my first effort!)

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    $\begingroup$ I'd look again at the details of my solution wrt the question if I were you. :) $\endgroup$
    – Donbhupi
    Commented Mar 2, 2015 at 13:32
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    $\begingroup$ squeamish - there are 121 competitors, so there are actually 17 competitors that finish behind Jessica in every event. That would put her top possible finish at 2nd (7*17 = 119). $\endgroup$
    – Duncan
    Commented Mar 2, 2015 at 13:34
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    $\begingroup$ @Donbhupi .... yes...? Wouldn't you? $\endgroup$
    – r3mainer
    Commented Mar 2, 2015 at 14:38
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    $\begingroup$ @Donbhupi Yes, you would. What would you prefer? That we say she placed second? But 103 people did better than her! $\endgroup$
    – KSmarts
    Commented Mar 2, 2015 at 15:06
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    $\begingroup$ This might raise some eyebrows. "No, of course I didn't rig the games - those seven 103-way ties happened by chance, I swear!" $\endgroup$
    – Milo Brandt
    Commented Mar 3, 2015 at 3:16

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