Official TSR Mathematical Society

Nice question;

Let A be the real sequence (a_1, a_2, a_3 ..) and define del(A) to be sequence (a_2 - a_1, a_3 - a_2, ... ). Suppose that all the terms of the sequence del(del(A)) are 1 and that a_(19) = a_(94) = 0. Find a_1

Sauce; Heinz - no wait AIME 1994

I'm getting

Yes, although not the method I'd apply in the AIME.

I would have gone. Sequences, differences, polynomial, roots, divide by 2: $\frac{1}{2}(1-19)(1-94)$

I think some people might appreciate it if you unpacked that comment slightly!

Show that if $p^2(x)-xq^2(x)=xr^2(x)$ then $p=q=r=0$ (not too hard, so don't do it if it's obvious for you)

That's not necessarily true

Couple of questions I came across earlier this week.

1) what is the sum of all 4 digit positive integer palindromes eg. 3223

2) In a school there are 1000 lockers and 1000 students. On the first day of school, the first student opens all the lockers. The second student then closes all the lockers with an even number. The third student will 'reverse', every third locker (if it is open, the student closes it. If it is closed, the student will open it). The fourth student reverses every fourth locker, and so on until all students have passed through. Which lockers will be open at the end?


The first is probably a fair bit harder than the second.

Take any one four digit palindrome;

1000a + 100b + 10b + a. a can be from the set {1,2,3,..9} and b from the set {0,1,2,...,9}.

If a = 1 then you are clearly summing

1000 + 1, 1000 + 100 + 10 + 1, ... , 1000 + 900 + 90 + 1

Let us call this sum S.

If a=2 we have a similar situation - but note that we only have to add 10(1000) + 10 to S to get his sum.

(2) Can you consider a much smaller scenario, say 10 lockers. Do you notice anything going one here say anything to do witht he divisors?

In know the locker one was on the oxford 2008 entry paper but where di you get the first?

I didn't know it was in the Oxford paper. It was also in MAO 1992.

The first one is from Mandelbrot 1.

Both are actually in the Art of Problem Solving Books.

Apparently it was also used as an interview question according to a random someone I sat next to at lunch in Keble :>

It is to do with squares.