MathDB

p10 Kathy has two positive real numbers,

a

and

b

. She mistakenly writes

\log (a + b) = \log (a) + \log( b),

but miraculously, she finds that for her combination of

a

and

b

, the equality holds. If

a = 2022b

, then

b = \frac{p}{q}

, for positive integers

p, q

where

gcd(p, q) = 1

. Find

p + q

.
p11 Points

X

and

Y

lie on sides

AB

and

BC

of triangle

ABC

, respectively. Ray

\overrightarrow{XY}

is extended to point

Z

such that

A, C

, and

Z

are collinear, in that order. If triangle

ABZ

is isosceles and triangle

CYZ

is equilateral, then the possible values of

\angle ZXB

lie in the interval

I = (a^o, b^o)

, such that

0 \le a, b \le 360

and such that

a

is as large as possible and

b

is as small as possible. Find

a + b

.
p12 Let

S = \{(a, b) | 0 \le a, b \le 3, a

and

b

are integers

\}

. In other words,

S

is the set of points in the coordinate plane with integer coordinates between

0

and

3

, inclusive. Prair selects four distinct points in

S

, for each selected point, she draws lines with slope

1

and slope

-1

passing through that point. Given that each point in

S

lies on at least one line Prair drew, how many ways could she have selected those four points?

Problem Statement