MathDB

Each problem in this section will depend on the previous one! The values

A, B, C

, and

D

refer to the answers to problems

1, 2, 3

, and

4

, respectively.
TR1. The number

2020

has three different prime factors. What is their sum?
TR2. Let

A

be the answer to the previous problem. Suppose

ABC

is a triangle with

AB = 81

BC = A

, and

\angle ABC = 90^o

. Let

D

be the midpoint of

BC

. The perimeter of

\vartriangle CAD

can be written as

x + y\sqrt{z}

, where

x, y

, and

z

are positive integers and

z

is not divisible by the square of any prime. What is

x + y

?
TR3. Let

B

the answer to the previous problem. What is the unique real value of

k

such that the parabola

y = Bx^2 + k

and the line

y = kx + B

are tangent?
TR4. Let

C

be the answer to the previous problem. How many ordered triples of positive integers

(a, b, c)

are there such that

gcd(a, b) = gcd(b, c) = 1

and

abc = C

?
TR5. Let

D

be the answer to the previous problem. Let

ABCD

be a square with side length

D

and circumcircle

\omega

. Denote points

C'

and

D'

as the reflections over line

AB

C

and

D

respectively. Let

P

and

Q

be the points on

\omega

, with

A

and

P

on opposite sides of line

BC

and

B

and

Q

on opposite sides of line

AD

, such that lines

C'P

and

D'Q

are both tangent to

\omega

. If the lines

AP

and

BQ

intersect at

T

, what is the area of

\vartriangle CDT

?
PS. You had better use hide for answers.

Problem Statement