North Dakota Mathematics Talent Search Questions (2000-2001)
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How many triples of positive integers (x, y, z) are there for which
28 x + 30 y + 31 z = 365?
- Over the top of a fence is placed a rope, with the same amount hanging
on each side. The rope weighs 1/3 pound per foot. On one end of the rope
hangs a monkey holding a banana; on the other end is a weight weighing the
same amount as the weight of the monkey.
The banana weighs 2 ounces per inch. The length of the rope (in feet) is
equal to the age of the monkey, and the weight of the monkey (in ounces) is
as much as the age of the monkey's mother. The combined ages of the monkey
and it's mother is thirty years.
The weight of the banana plus one half the weight of the monkey is one
quarter as much as the sum of the weights of the weight and the rope, where
all weights are in the same units.
The monkey's mother is one half as old as the monkey will be when it is
three times as old as it's mother was when she was one half as old as the
monkey will be when it is twice as old as it is now.
How long is the banana?
- Suppose that P and Q are two distinct positive integers, and consider
the set of whole number combinations of P and Q. For example 0, 2 P, and 3
P + 5 Q are all examples of whole number combinations of P and Q. Let S(P,
Q) denote the set of all possible whole number combinations of P and Q.
Then, for example, S(2, 3) = {0, 2, 3, 4, ... } is the set of all whole
numbers not equal to 1, while S(2, 6) = {0, 2, 4, 6, ... } is the set of all
even whole numbers.
Determine, as a function of N, how many whole numbers are NOT
in S(N, N+1).
- We place five identical spheres in a rectangular box as shown below.
The bottom of the box is a 10 cm by 10 cm square. The spheres are packed
as tight as possible, with the center sphere touching the other four, and
all five spheres touching the top and bottom of the box. How tall is the
box? (Give both an exact answer and a decimal approximation)
- Find two positive integers M and N so that:
(i) M/N = 1/2, and
(ii) the union of the digits in the decimal (base 10) representation of M
and N consists of {1, 2, ..., 9} with no repeats.
- In a normal game of Tic-Tac-Toe, player X wins if she places three X's
in the same row, in the same column, or in a diagonal, as shown in figure
(a) below. In a game of "wrap-around" Tic-Tac-Toe, a player can also wrap
around the edges of the 3x3 grid to complete a winning three-in-a-row, as
indicated in figure (b) below. Therefore, we can think of the top three
squares in the grid as being directly below the bottom three squares and
the three leftmost squares as being directly to the right of the three
rightmost squares. If in a normal game of Tic-Tac-Toe, there are eight
ways for player X to win, how many ways are there in wrap-around
Tic-Tac-Toe?
- An NDSU mathematics major paddled six miles upstream on the Red River
of the North, at which point his hat fell into the river. Without stopping,
he continued to paddle upstream at the same rate for two more hours. Then
he turned and paddled back to the starting point, arriving at exactly the
same time as his hat, which had floated downstream after falling off. How
fast was the river flowing?
- Suppose that I have a whole bunch of poker chips, each one with a
different positive integer written on it. Before eating my turkey dinner
on Thanksgiving Day, at one minute before five, I put nine chips, labelled
1 through 9, in a large sack. At a half a minute before five, I add ninety
chips, labelled 10 through 99, to the sack and immediately take out chip
number 1. At a quarter of a minute before five, I place nine hundred chips
labelled 100 through 999 in the sack and remove chip number 2. Then one
eighth of a minute before five I add nine thousand chips labelled 1000 to
9999 and remove chip number 3. Assuming that I can continue to do this
(so I can move arbitrarily fast) and I don't run out of chips or room in
the sack, how many chips will there be in the sack when I sit down to eat
dinner at five?
- In a game of American Football, most of the points scored come from field
goals and touchdowns. A field goal is worth 3 points. A touchdown is
worth 6 points plus the opportunity to score an extra point; thus most
touchdowns are worth 7 points. What is the largest number that cannot be a
valid score in a football game in which all points are scored from
touchdowns and field goals (i.e. no safeties or two-point conversions)?
- Nine coins look alike but one is a counterfeit and weighs less than
the others. Describe how, using only a balance, you can find the fake in
only two weighings.
- A square is inscribed in a circle which is circumscribed by an
equilateral triangle, as indicated by the figure below. How many times
larger is the area of the triangle as that of the square?
- A geography class at NDSU contains 8 men and 7 women. If the professor
selects 3 people at random to work on a presentation on the economic
effects in Bolivia arising from its being a landlocked country, what is
the probability that the number of men selected exceeds the number of women
selected?
- What is the smallest whole number N whose decimal expansion ends in a "3"
and has the property that when we move that "3" to the front of the decimal
expansion and shift all of the other digits down by one position we get 3N?
- Next year, 2002, is a palindrome year because it reads the same forwards and
back. The next palindrome year after that will be 2112. 2002 and 2112 are
a whole 110 years apart. After the year 1000, what is the shortest period
of time separating two palindrome years?
- Suppose we made a rope lasso long enough to encircle the entire world at the
equator. How much longer would it have to be so that the lasso could be
loosened up and lifted up one kilometer off the face of the earth?
- The Prince of Bombay has a beautiful garden in the middle of which sits the
largest mango tree imaginable. This tree gives the biggest, best tasting
mangoes in the world, each one weighing about two pounds! To keep the tree
and its fruit all to himself, the Prince builds seven concentric walls
surrounding the garden. Each wall has a single gate with a eunuch guarding
the gate. In order to get to the mango tree, you would have to go past all
seven guards.
One day a small man approaches the outside guard and proposes the following
deal: If the guard lets him past the gate and he can get to the mango tree
then the man promises to give the guard one less than half of the mangoes
he is carrying on his way out. The outside guard accepts the deal and lets
him through. The man then goes up to the second guard, makes the same deal
with him and gets past the second gate. He does the same bit of convincing
five more times and eventually gets to the mango tree.
If no mangoes are to be split in half, how many mangoes must the small man
take from the tree so that he can make good on his seven promises?
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