1998 TRINITY MATH TRIATHLON

1.   Students fall asleep in a class at the rate of one student for every 30 seconds.  At the tenth minute, the sleeping students start waking at the rate of one student for every minute. If the class has 30 students, how long after class begins will all the students be asleep?  Note: Students may fall asleep more than once, and at the start of class all are assumed to be awake.

2.  A lucky year is one in which at least one date, when written in the form month/day/year has the following property: The product of the month times the day equals the last two digits of the year.  For example, 1956 is a lucky year because it has the date 7/8/56 and 7×8=56.  Decide which of the following years are lucky and demonstrate how: 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999.

3.  How many different triangles are contained in this figure? 

4.  Twelve-hour digital clocks can display how many numbers that are perfect squares?  Example 8:41 gives 841 = 29²;  but 13² = 169 has no corresponding time.

5. Consider the twelve cards attached.  (A small image of the twelve cards with labeling is below.)  Each card has a color, a shape, a quantity, and a shading.  A set is three cards on which each feature is either the same on all of the cards or different on all of the cards.  For example, cards F H J form a set, but cards F H M do NOT form a set.  Find the five remaining sets.  Note that the leters are not a feature, just a label.