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### Card 129

Lines *DE* and *BC* are parallel. Points *A, D, B* are collinear, and angles *ADE* and *ABC* are equal. Points *A, E, C* are collinear in that order, and segments *AE* and *EC* are equal.

*What is the area of quadrilateral BCED divided by that of triangle EAD?*

### Card 94

Right triangle *ABC* has short leg *AC* of length 1/2 and hypotenuse *BC* of length 1. Circle *L* is tangent to *AB* at *B* and crosses *BC* at *D*.

*What is the measure of the longer arc between B and D on L?*

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## Solutions to week 64

**Skimpy Schedule**. There are three different rooms, so clearly at least three students are needed. Because there are only four time slots and a “no three in a row” rule for competing in events, each student can compete in at most three events out of the twelve, so actually at least four students are needed. However, the rule is more restrictive; because a competitor cannot attend three events in a row, in order to attend three events, s/he must sit out one of the two middle events. Thus if there were only four competitors, they could only cover four out of the the six events in the two middle time slots. Hence, at least **five competitors** are needed, and it can be done with five, if the students competing in each of the four time slots are arranged as ABC, ABD, CDE, ABC (among many possibilities).

**Passing Pennies**. In order to achieve the minimum pennies left, we want the game to go on as long as possible with as many odd pennies occurring as possible. Since the game ends when the pennies are evenly distributed among the three players, we should therefore look at the most uneven initial distribution of pennies possible. In other words, what happens when one student starts with all of the pennies? Then the game goes as follows:

26 0 0 -> pass -> 13 13 0 -> give back -> 12 12 0 -> pass -> 6 12 6 -> pass -> 6 9 9 -> give back -> 6 8 8 -> pass -> 7 7 8 -> give back -> 6 6 8 -> pass -> 7 6 7 -> give back -> 6 6 6 (done)

Therefore, each student will end up with at least **six pennies**.