North Dakota Mathematics Talent Search 2004-2005
PROBLEM SET 3
(Due 05/01/2005)
- Show that the integral part of
(2 + 31/2)N is odd
for all positive integers N.
- There is a tennis tournament with 937 players. The player
who wins a game advances further, while the loser leaves the
competition. How many games are required to complete the
tournament?
- In front of you is a glass of water and a glass of juice.
Their volumes are identical. You take a teaspoon, put it
in the glass of juice, take a full spoon, and put it in
the water. After briefly stirring up the mixture, you put
a full teaspoon of the mixture back into the glass of juice.
The question is which amount is greater: the amount of juice
in the water or the amount of water in the juice?
- The minute hand of an analog clock covers the hour hand at noon.
At what time will both hands overlap again? (We need to
determine the nearest such moment.)
- There are two cities, A and B, that are 400 km
apart. Simultaneously, a train leaves city A going
toward city B with a constant speed of 40 km/hour,
and another train leaves city B going toward city A
with a constant speed of 60 km/hour. Also, at the same time,
a busy bee, which was resting at the front of the first train,
starts an interesting trip. It flies with a constant speed
of 75 km/hour toward the second train, and as soon as it touches
the second train, it reverses the direction of its flight. The
bee does this every time it meets a train. If the bee flies
this way until two trains meet, what is the total distance
traveled by the bee?
|
|
