North Dakota Mathematics Talent Search Questions (2001-2002)
-
It takes 8 hours less to fill a pond using a large pipe than it takes
using a smaller pipe. If, when used together, they fill the pond in 3
hours, how long would it take each to fill it alone?
- Let A1, A2, A3, ... , An,
represent an arbitrary arrangement of the numbers 1, 2, 3, ... , n. Prove
that if n is odd, the product (A1 - 1) (A2 - 2)
(A3 - 3) ... (An - n) is an even number.
- How many positive integers of n digits exist such that each digit is
1, 2, or 3? How many of these contain all three of the digits 1, 2, and 3
at least once?
- The lengths of sides CB and CA of triangle ABC are a and b,
respectively, and the angle between them measures 120o. Express
the length of the bisector to angle C in terms of a and b.
-
A single square is removed from a 2N by 2N
checkerboard. Show that (regardless of which square is missing) the
checkerboard can be tiled with non-overlapping pieces of the shape below.
(Each square is the size of one of the squares of the checkerboard and the
pieces can be rotated).
- Start with a randomly chosen positive integer and take its square
root. Double the result and take the square root again. Continue
doubling and taking square roots indefinitely. What value are you getting
closer and closer to? Explain why.
- I was about to get $0.83 in change at the Math Emporium last night
when there was a blackout and the entire store became pitch black. At the
moment when the lights went out, the cashier was handing me eight coins
(quarters, dimes, nickels, and pennies). How do I know he gave me the
wrong amount of change?
- For what values of c does the formula
x3 + c x2 + c x + 1 = 0
have exactly one solution?
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