- #1
Doom of Doom
- 85
- 0
Did anyone else toil away for six hours yesterday at the Putnam exam?
I'm fairly confident that I got a couple.
________
A1: Find a where y=a x^2 +a x + \frac{1}{24} and x=a y^2 +a y + \frac{1}{24} are tangent.
This was easy, once you recognize that the curves could only intersect (i.e. be tangent) along the y=x line.
_____
A2: Find the least possible area of a convex set in the plane that intersects both branches of the hyperbolas xy=1 and xy=-1. (A set S is called convex if for any two points in S the line segment connecting them is contained in S)
It seems obvious to me that this area would be 4, but I spent quite a bit of time on this problem and wasn't able to turn in a good solution for it.
______
A3: Take the integers 1,2,...,(3k+1) and write them down in a random order. What is the probability that, as you write them down, there is no time when the sum of all of the numbers written up to that point is divisible by 3?
If you take all of the numbers and mod them by 3, you get: k+1 ones, k twos and k zeros. Then, if you ignore the zeros (as long as you don't write one in the first position), the only way you can write the one's and two's is like this (and this is easy to show):
1,1,2,1,2,1,2,...,1,2
So, you can count the number of ways that you can write that out, then count the number of ways that you can insert the zeros into that sequence, and divide out by (3k+1)!.
_______
Those were the problems I got in the first session. The only problem I solved in the second session was B1.
_______
B1: If f(n) is a polynomial with positive integer coefficients, prove that the only case when f(n) divides f(f(n)+1) is when n=1.
This was very easy. All you have to do is mod by f(n), and you get f(n) (mod f(1)) =0 which can only occur when n=1 (because all of the coefficients are positive).
____
B3: Find x_{2007} where x_0=1 and x_{n+1}=3x_n+\left\lfloor x_n \sqrt{5}\right\rfloor
I didn't turn in a solution for this, and I was never able to get this in closed form, but here is what I reduced it to:
x_n=(3+\sqrt{5})^n-(\sqrt{5}-2)\sum_{i=0}^{n-1} x_i
_____
That's what I turned in. So, I hope that I get 30 pts. How did everyone else do?
I'm fairly confident that I got a couple.
________
A1: Find a where y=a x^2 +a x + \frac{1}{24} and x=a y^2 +a y + \frac{1}{24} are tangent.
This was easy, once you recognize that the curves could only intersect (i.e. be tangent) along the y=x line.
_____
A2: Find the least possible area of a convex set in the plane that intersects both branches of the hyperbolas xy=1 and xy=-1. (A set S is called convex if for any two points in S the line segment connecting them is contained in S)
It seems obvious to me that this area would be 4, but I spent quite a bit of time on this problem and wasn't able to turn in a good solution for it.
______
A3: Take the integers 1,2,...,(3k+1) and write them down in a random order. What is the probability that, as you write them down, there is no time when the sum of all of the numbers written up to that point is divisible by 3?
If you take all of the numbers and mod them by 3, you get: k+1 ones, k twos and k zeros. Then, if you ignore the zeros (as long as you don't write one in the first position), the only way you can write the one's and two's is like this (and this is easy to show):
1,1,2,1,2,1,2,...,1,2
So, you can count the number of ways that you can write that out, then count the number of ways that you can insert the zeros into that sequence, and divide out by (3k+1)!.
_______
Those were the problems I got in the first session. The only problem I solved in the second session was B1.
_______
B1: If f(n) is a polynomial with positive integer coefficients, prove that the only case when f(n) divides f(f(n)+1) is when n=1.
This was very easy. All you have to do is mod by f(n), and you get f(n) (mod f(1)) =0 which can only occur when n=1 (because all of the coefficients are positive).
____
B3: Find x_{2007} where x_0=1 and x_{n+1}=3x_n+\left\lfloor x_n \sqrt{5}\right\rfloor
I didn't turn in a solution for this, and I was never able to get this in closed form, but here is what I reduced it to:
x_n=(3+\sqrt{5})^n-(\sqrt{5}-2)\sum_{i=0}^{n-1} x_i
_____
That's what I turned in. So, I hope that I get 30 pts. How did everyone else do?