Competition problem — Algebra — Maths Olympiad Prep

Library / /352 of 520

Algebra Difficulty 5.8 AIME, harder Prove it

5. Let $a_{1}, a_{2}, \ldots, a_{2021}, b_{1}, b_{2}, \ldots, b_{2021}$ be pairwise distinct natural numbers. Consider the graphs of functions of the form

$y=\frac{a_{i}}{x+b_{i}}$

(2021 functions in total). Can it happen that the abscissas of all intersection points of these graphs are integers?

Solution

# Solution.

It can. Consider some functions of the specified form with pairwise distinct coefficients. Find the abscissa of the intersection point. Solve the equation $\frac{a_{i}}{x+b_{i}}=\frac{a_{j}}{x+b_{j}}$ , we get $x=\frac{b_{i} a_{j}-a_{i} b_{j}}{a_{i}-a_{j}}$ . Let $N-$ be the LCM of all numbers $a_{i}-a_{j}$ . Multiply all coefficients $a_{i}$ and $b_{i}$ by $N$ . Then all numerators of the abscissas of the intersection points will be divisible by $N^{2}$ , and the denominators will be equal to $N\left(a_{i}-a_{j}\right)$ , which is a divisor of $N^{2}$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.