p16 Madelyn is being paid $50/hour to find useful Non-Functional Trios, where a Non-Functional Trio is defined as an ordered triple of distinct real numbers (a,b,c), and a Non- Functional Trio is useful if (a,b), (b,c), and (c,a) are collinear in the Cartesian plane. Currently, she’s working on the case a+b+c=2022. Find the number of useful Non-Functional Trios (a,b,c) such that a+b+c=2022.
p17 Let p(x)=x2−k, where k is an integer strictly less than 250. Find the largest possible value of k such that there exist distinct integers a,b with p(a)=b and p(b)=a.
p18 Let ABC be a triangle with orthocenter H and circumcircle Γ such that AB=13, BC=14, and CA=15. BH and CH meet Γ again at points D and E, respectively, and DE meets AB and AC at F and G, respectively. The circumcircles of triangles ABG and ACF meet BC again at points P and Q. If PQ can be expressed as ba for positive integers a,b with gcd(a,b)=1, find a+b. number theoryalgebrageometryYale