MathDB

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) = x^2 - 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

\Gamma

such that

AB = 13

BC = 14

, and

CA = 15

BH

and

CH

meet

\Gamma

again at points

D

and

E

, respectively, and

DE

meets

AB

and

AC

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

\frac{a}{b}

for positive integers

a, b

with

gcd (a, b) = 1

, find

a + b

Problem Statement