North Dakota Mathematics Talent Search Questions (1999-2000)
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In a lottery the tickets are numbered and sold consecutively numbered with six-digit numbers from 000,000 to 999,999 and then starting over again. Special $100 prizes are
awarded for tickets in which the first three digits are the same and in the same order as the last three digits. How many tickets must one buy to be assured of getting one of these
$100 prizes?
- One hundred students participated in a mathematics competition consisting of four problems. No student solved every problem. The first problem was solved by exactly 90
students; the second by exactly 80, the third by exactly 70, and the fourth by exactly 60 students. A prize was awarded only to each student who solved both the third and fourth
problems. How many prizes were awarded?
- Points O, A, B, and C are located on a number line at coordinates 0, 8, 12, and 26 respectively. Let P be a point not on the number line. Let Q be the midpoint of
segment PA. Let R be the midpoint of segment QB. Let S be the midpoint of segment PC. Find the coordinate of the point where the line SR intersects the number line.
- Winnie-the-Pooh and Rabbit took a bag of 1999 candies to play a mathematical game. Each of them in turn takes 1, 2, or 3 pieces of candy from the bag. The player who takes
the last candy from the bag is declared the winner. Can Winnie do anything to be certain of winning this game? Describe what or explain why not.
- Bill Gates has exactly half a million bills, each of which is one of
the following denominations: $1, $10, $100, or $1,000. Is it possible that
these bills add up to exactly one million dollars.
- Given the sequence t1, t2, t3, ...
with 2 Sk + 3 tk = 10 for each k greater than 1.
(NOTE: Sk denotes the sum of the first k terms of the sequence
t1, t2, t3, ... ).
(a) Show that the sequence is a convergent geometric sequence.
(b) For what values of k does Sk differ less than 0.0001 from
its limit?
- Describe how to cut a parallelogram along a straight line through its
center so that the two pieces can be rearranged to make a rhombus.
- Let a and b be positive integers. Show that the square root of 2
always lies between the two fractions a/b and (a + 2 b)/(a + b).
- For which integer values of x is | 12 - x2 | =
x2 - 12 ?
- (a) The sum of the three consecutive integers 1, 2, 3 is equal to
their product: 1 + 2 + 3 = 1 * 2 * 3 = 6. Find all other sets of three
consecutive integers with the same property (i.e., that their sum is equal
to their product).
(b) Are there any sets of four consecutive integers with the same
property? How many sets of five consecutive integers have that property?
Be sure to support your answers with explanations.
- Consider the trapezoid ABCD with right angles at A and B and with AC =
AD. Let S be the point of intersection of the two diagonals, AC and BD.
Prove that the circle with center S and radius SC is tangent to the side
AB.
- The Fargo Association of Hospitals is planning on building a
state-of-the-art facility for its blood bank. Part of the planning for
this new facility is to determine the best location for the blood bank.
There are three hospitals which will be requiring regular deliveries from
the blood bank. Their location and the estimated number of deliveries
required each week is given by the following table:
| Hospital |
Location |
Number of trips per week |
| A |
(10, 90) |
2 |
| B |
(70, 50) |
4 |
| C |
(40, 20) |
5 |
Assume that the streets in the city of Fargo are set up like a grid, so if
you drive between two points with coordinates (a, b) and (c, d), the
distance traveled is D = | a - c | + | b - d |. Also assume that the
delivery trucks go to only one hospital and back each trip. What are the
coordinates for the optimal location of the new blood bank?
- (a) Find two positive integers, M with exactly four divisors and N
with exactly five divisors, that have exactly two common divisors. (Note:
The common divisors of 4 and 6 are 1 and 2).
(b) Give a complete explanation of how to find all such pairs of positive
integers.
- An NDSU mathematics major paddled six miles upstream on the Red River
of the north, at which point her hat fell into the river. Without stopping,
she continued to paddle upstream at the same rate for two more hours. Then
she turned and paddled back to the starting point, arriving at exactly the
same time as her hat, which had floated downstream after falling off. How
fast was the river flowing?
- Given the function f(x) = 3 x2 - 4 (m + 1) x + 4 m:
(a) Show that the equation f(x) = 0 has two distinct roots x1
x2, for every value of m.
(b) Find the value of m for which | x1 - x2 | is
minimal.
- Prove that if
 ,
then  .
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