2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 9
Problem
Find the number of -tuples with and positive integers, such that and have integer roots
Solution
Let , be the roots of the first quadratic; , the roots of the second quadratic; , the roots of the third; and , the roots of the fourth quadratic ( for ). Using this, we can write the the four equations in terms of (for ): By Vieta's (or expanding out the binomials), we find that Adding all the equations together, Rearrainging, Factoring with SFFT, By inspection, we find that the only solution that satisfies ALL the above equations is , which gives solution for (that is ).
See also
2018 UNM-PNM Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNM-PNM Problems and Solutions |