Fall 2005
Week 1’s
Problem: Some Addition
Since 2005 = 1002 + 1003, the number 2005 can be written as the sum of two consecutive positive integers. For 5 points, how many ways can 2005 be written as the sum of (more than two) consecutive positive integers? For another 5 points, how many ways can 2005 be written as the sum of consecutive integers if the numbers can be both positive and negative?
Week 2’s
Problem: A Little Algebra
Are there any pairs (x, y) of positive integers that satisfy x2 – y2 = 2005? If so, how many different solution pairs are there?
Week 3’s
Problem: Points in the Plane
Consider all line segments of length 4 with one endpoint on the line y = x and the other endpoint on the line y = 2x. Find an equation for the curve that consists of all of the midpoints of the line segments of length 4.
Week 4’s
Problem: Polygons in Polygons
Specifically, find the largest regular hexagon (all sides and angles equal) that you can put inside a square of side length 1.
Week 5’s Problem: More Polygons
Consider the set of all regular n-gons (polygons with all sides and angles equal) of some fixed area A. Is there an n-gon in the set whose perimeter is smaller than the perimeter of every other polygon in the set? If so, give the value of n with justification. If not, give a careful argument why there is no smallest perimeter in the set.
Week 6’s
Problem: That’s Some Sum!
Let [x] denote the greatest integer function, that is, the function that returns the greatest integer less than or equal to x.
For n a positive integer greater than 0, find a simpler expression for the following sum:
[(n + 1)/2] + [(n + 2)/4] + [(n + 4)/8] + [(n + 8)/16] + …
You can get four points by identifying what the sum is and the other six points by providing a convincing argument that your formula is correct.
Week 7’s
Problem: Guess what I’m thinking
Alice, Bob, and Chris are playing
number games one day. Chris says to the other two: "I am thinking of two
integers x and y where 3
≤ x ≤ y ≤ 97. I'll tell
Bob: "That was true, Alice, but now I do."
Is it really possible for Alice and Bob to determine x and y? If so, what are x and y? If not, why is it impossible to tell?
Week 8’s
Problem: A Positive Polynomial
We’ll end this semester with a two part problem with each part worth five points.
1) Let f (x) be a polynomial of degree 2 that is greater than or equal to 0 for all values of x. Show that f” (x) + f’ (x) + f (x) is also greater than or equal to 0 for all values of x.
2) Let f (x) be a polynomial of degree n that is greater than or equal to 0 for all values of x. Show that f (n) (x) + … + f” (x) + f’ (x) + f (x) is also greater than or equal to 0 for all values of x. (In the sum, f (n) (x) means the nth derivative.)
Spring 2006
Week 1’s Problem: What’s the difference?
Take a four digit number, reverse its digits, and look at the absolute value of the difference of the two, like so: | 3467 – 7643 | = 4176.
Can you ever get 2006 as the answer?
Week 2’s Problem: Lots ‘o Letters
Take the collection of “words” made from the alphabet {a,b,c}. Words may be transformed into other words using the following bidirectional rules:
1. a « ab
2. bc « ba
3. ac « cab
4. aba « ac
5. ab « ba
Example: Start with aabab. Then : aabab = aa(ba)b « aa(bc)b = a(ab)cb « a(a)cb = a(ac)b« a(cab)b
So aabab can be transformed into acabb.
a) Can cbaba be transformed into aca?
b) Can baca be transformed into baa?
Week 3’s Problem: Wild Cards
A stack of 2n cards (labeled 1 through 2n, top to bottom) are sitting on a desk in a neat pile. We reorder the pile in a very special way, as follows. First, take the top n cards off and put them in a pile (call it pile A) next to the remaining n cards (which we'll call pile B). Then, form the new, reordered pile from bottom to top: First take the top card from pile B and put it down on the table (this is the (n+1)st card from the original pile). Now take the top card from pile A and put it on top of that card. Thus, after two steps, we have two cards in our new pile: card n+1 (on the bottom) and card 1 (above card n+1). Continue in this manner, alternately placing one card from pile B and then one from pile A to create the new, reordered pile.
Example: If n = 3, then the 6 cards will be reordered as 3 6 2 5 1 4, where 3 is the new top card.
A card may be in the same position in the reordered pile as it occupied in the original pile. Call such a card lucky.
a) (3pts) If n = 2006, are there lucky cards? If so, what card numbers are they?
b) (3pts) Are there any values of n for which card number 2006 in the stack is lucky? If so, what are they? If not, why not?
c) (4pts) What is the maximum number of lucky cards there can be in a stack? For which value(s) of n does this situation occur?
Week 4’s Problem: It’s Just a Volume
Find the volume of the region in space whose points (x, y, z) satisfy the inequality | x | + | y | + | z | + | x + y + z | ≤ 2.
Week 5’s Problem: Doing Things in Sequence
Let a, b, and c be the cube roots of three distinct prime numbers. Can a, b, and c ever be three terms of an arithmetic progression?
Notes:
1) An arithmetic progression is of the form m, m + n, m + 2n, … , m + kn,
2) The numbers a, b, and c, do not need to be consecutive terms in the progression
Week 6’s Problem: Boxed In
The diagram below shows ten rectangles arranged in something of a staircase pattern. The four values A, B, C, and D are vertical distances, whereas a, b, c, and d are horizontal distances. These eight distances are allowed to take on any real value, with the restriction that each of the ten rectangles must have area that is an integer, and the ten rectangle areas must all be distinct.
As an example, if A = 1/5, then the value of c could be 10, but cannot be 7. If A = 1/5 and c = 10, then only the rectangle with side lengths A and c can have area 2.
Week 7’s Problem: Crossed Wires
Ella the electrician has been assigned to label wires that run through a pipe which passes under a busy street. The street runs North-South and the pipe runs East-West. There are 210 identical wires running through the pipe. At the West end of the pipe, each wire is labeled 1, 2, 3, ...., 210, but there are no such labels on the ends of the wires which emerge at the East end. Ella's job is to correctly label these wires. Ella plans to connect some or all of the wires together in groups at the West end, cross the street, and run current through the wires at the East end to see which wires are connected to each other. She will then cross the street and repeat the process. She will repeat the process as many times as necessary to correctly label all of the wires on the East side. She wants to accomplish this labeling while crossing the street as few times as possible. How does she proceed?
Week 8’s Problem: The Triangle Inequality?
Not exactly, but there is a triangle involved.
Find all real values of x, y, and z so that the inequality
xA2 + yB2 + zC 2 ≤ 0
is satisfied, for all values of A, B, and C that are the sides of a triangle. (That is, the (x, y, z) values should work simultaneously for all A, B, and C that are the sides of a triangle.)