APMO #1 1997

[stupidityismygam]

Given:

[unparseable or potentially dangerous latex formula]

where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. [unparseable or potentially dangerous latex formula] for [unparseable or potentially dangerous latex formula] Prove that [unparseable or potentially dangerous latex formula]

[bradp]

yay number sense problems
The sum of [unparseable or potentially dangerous latex formula] reciprocals of consecutive triangular numbers starting with the [unparseable or potentially dangerous latex formula] triangular number is

[unparseable or potentially dangerous latex formula]

so each term is the reciprocal of that formula where n = 1 and x=1,2,3...1996

put it all into the calculator [unparseable or potentially dangerous latex formula]

[stupidityismygam]

[bradp]

how do you do sigma notation with LaTex?

/*The [unparseable or potentially dangerous latex formula] term is
[unparseable or potentially dangerous latex formula]
so some terms are....
[unparseable or potentially dangerous latex formula]*/

The series asymptotically reaches 1/2
If each term was [unparseable or potentially dangerous latex formula], the sum would be 1996 * 1/2 = 998.

Looking at the terms, there is clearly enough left over after subtracting out that 1/2 (as seen below) to get the extra 3 that we need!

[unparseable or potentially dangerous latex formula]
That looks like similar to the 'harmonic sequences' that have been showing up on the math tests this year.. I need to find a partial sum formula for it and i can then finish this beast of a problem

[zefuri]

and clearly calculator is not number sense...

[stupidityismygam]

[bradp]

[stupidityismygam]

try it without a program/calc :) :)

because APMO laughs at peopel who use calculators

[bradp]

I reciprocate by laughing at APMO.

[stupidityismygam]

APMO laughs at number sense...

it actually requires brain power (as opposed to UIL, no offense of course)

[zefuri]

does anyone know how to do harmonic series with just even numbers?

[Kurt]

What does APMO stand for? I've never heard of that one (or almost any of the ones that have popped up in the forums today).

Anyway, I'll give the problem a shot:

You have the sum of the reciprocal of triangular numbers:

Since [unparseable or potentially dangerous latex formula]

Decomposing into partial fractions, [unparseable or potentially dangerous latex formula]

So [unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]. It's obvious that the inner sums will cancel with each other, and so all that is left is: [unparseable or potentially dangerous latex formula] for the sum of k triangular inverse numbers [unparseable or potentially dangerous latex formula]

So [unparseable or potentially dangerous latex formula].

And so [unparseable or potentially dangerous latex formula].

Which is [unparseable or potentially dangerous latex formula].

Now using a handy trick for powers of 2 and reciprocals: [unparseable or potentially dangerous latex formula].
This is true by AM>HM, (or in this case, 1/HM > 1/AM)

We have 9 such cases here (from [unparseable or potentially dangerous latex formula]to [unparseable or potentially dangerous latex formula])

And of course we have a couple of terms before that ([unparseable or potentially dangerous latex formula]).

So total we have [unparseable or potentially dangerous latex formula].

However, for the part of the sum that is the sum of the reciprocals, that is just 1024 of the 1996 terms...So it is obviously greater than 1001.

[stupidityismygam]

APMO=Asian Pacific Mathematical Olympiad
GMO=Greece
IRMO=Indian Regional
USAMO=United States of America