Difference between revisions of "1992 OIM Problems/Problem 2"

Revision as of 11:29, 17 December 2023

Given the collection of $n$ positive real numbers $a_1 < a_2 < a_3 < \cdots < a_n$ and the function:

$f(x) = \frac{a_1}{x+a_1}+\frac{a_2}{x+a_2}+\cdots +\frac{a_n}{x+a_n}$

Determine the sum of the lengths of the intervals, disjoint two by two, formed by all $f(x) = 1$ .

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Since $a_1 < a_2 < a_3 < \cdots < a_n$ , we can plot $f(x)$ to visualize what we're looking for:

Notice that the intervals will be: $I_1=r_1-(-a_1), I_2=r_2-(-a_2), \cdots , I_n=r_n-(-a_n)$

Thus the sum of the intervals will be: $\sum_{i}^{}\left( r_i+a_i \right)$

Now we set $f(x)=1$ :

$f(x)=\frac{\sum_{j\ne i}^{}\left( a_i \prod_{j}^{}\left(x+a_j \right)\right)}{\prod_{i}^{}\left( x+a_i\right)}=1$

And solve for zero:

$\sum_{j \ne i}^{}\left( a_i \prod_{j}^{}\left(x+a_j \right)\right)-\prod_{i}^{}\left( x+a_i\right)=0$

$\left( x^n+\sum_{i}^{}a_ix^{n+1}+K_{n-2}x^{n-2}+\cdots+K_1x+K_0\right)-\left( \sum_{i}^{}a_ix^{n+1}+L_{n-2}x^{n-2}+\cdots+L_1x+L_0\right)=0$

Note. I actually competed at this event in Venezuela when I was in High School representing Puerto Rico. I got a ZERO on this one because I didn't even know what was I supposed to do, nor did I know what the sum of the lengths of the intervals, disjoint two by two meant. A decade ago I finally solved it but now I don't remember how. I will attempt to solve this one later.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

@@ Line 24: / Line 24: @@
 <math>\sum_{j \ne i}^{}\left( a_i \prod_{j}^{}\left(x+a_j  \right)\right)-\prod_{i}^{}\left(  x+a_i\right)=0</math>
+<math>\left( x^n+\sum_{i}^{}a_ix^{n+1}+K_{n-2}x^{n-2}+\cdots+K_1x+K_0\right)-\left( \sum_{i}^{}a_ix^{n+1}+L_{n-2}x^{n-2}+\cdots+L_1x+L_0\right)=0</math>
 * Note.  I actually competed at this event in Venezuela when I was in High School representing Puerto Rico.  I got a ZERO on this one because I didn't even know what was I supposed to do, nor did I know what the sum of the lengths of the intervals, disjoint two by two meant.  A decade ago I finally solved it but now I don't remember how.  I will attempt to solve this one later.