Here is something a little more light-hearted than my usual flux problems, in the spirit of the holidays.

Romeo and Juliet have a date at a lake whose shore line is given by $x^2+y^2=1$ , with distances measured in kilometers. Due to an unfortunate misunderstanding, they find themselves at the points $(1,0)$ and $(-1,0)$ , respectively, as they notice their error. Naturally, they wish to meet up as soon as possible, by all means necessary. They will each run along the lake for a while, northwards, towards the point $(0,1)$ , then jump into the water and swim towards each other (they have the option to swim all the way to the center of the lake or run all the way to the point $(0,1)$ , of course). They can both run at a speed of 10 km/h and swim at 5 km/h. How many meters should each of them run? Round your answer to the nearest integer.

None of the others
1047
0
524
785
1571

**
This section requires Javascript.
**

You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.

Comrade Huan has submitted a fine mathematical solution. The point of this problem, besides its (small) entertainment value, is that a critical point does not always produce the desired extremum; one needs to "check."

One can also reason this problem out as follows:

Since the two lovers swim half as fast as they run, swimming will save them time only if they can cut the distance in less than half. But any arc to a semicircle is less than twice as long as the corresponding chord, so that swimming would be a waste of time. The lovers are better off running all the way to $(0,1)$ , travelling $\frac{\pi}{2}$ km or about $\boxed{1571}$ meters.