North Dakota Mathematics Talent Search 2004-2005
SOLUTION SET 3
(Posted on 05/01/2005)
- Note that
(2 + 31/2)N +
(2 - 31/2)N is even for all
positive integers N. Indeed, all of the components
with 31/2 raised to an odd power will
cancel out (having opposite signs in the sum of the two
expressions), and even powers of 31/2
are either multiplied by 2 in some power, or added
together (e.g. 3N+3N =
2*3N for even N). Thus, our
(2 + 31/2)N happens to be
always equal to an even number minus
(2 - 31/2)N . It is also easy
to see that (2 - 31/2)N
is always a fraction from the range (0,1). Therefore
(2 + 31/2)N is always equal
to an even number minus a fraction, and so it is always odd indeed.
- Note that every game of the tournament eliminates a single
player. During the tournament of
N players we
have to eliminate N-1 of them to get a winner,
therefore we need N-1 games. So, for 937 players,
we need 936 games.
- Note that on the way back from the mixture to the juice, the
amount of water in the teaspoon has to be exactly the same
as the amount of juice we are leaving behind, because we just
transported one teaspoon of juice. Therefore the amount of
water in the juice ends up to be exactly the same as the amount
of juice in the water.
- Note that clock hands overlap 11 times every 12 hours. Since
they overlap at regular intervals, this happens every
1+1/11 hours.
- The two trains will meet after four hours since both of them
cover 40+60=100 km every hour of the total distance of 400 km.
In turn, the bee flies continuously for four hours, and at a
constant speed of 75 km/hour, therefore the total distance
traveled by the bee is 300 km.
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