MathDB

Guts Round / Set 7
p19. Let

a

be the answer to Problem 19,

b

be the answer to Problem 20, and

c

be the answer to Problem 21.
Compute the real value of

a

such that

\sqrt{a(101b + 1)} - 1 = \sqrt{b(c - 1)}+ 10\sqrt{(a - c)b}.

p20. Let

a

be the answer to Problem 19,

b

be the answer to Problem 20, and

c

be the answer to Problem 21.
For some triangle

\vartriangle ABC

, let

\omega

and

\omega_A

be the incircle and

A

-excircle with centers

I

and

I_A

, respectively. Suppose

AC

is tangent to

\omega

and

\omega_A

E

and

E'

, respectively, and

AB

is tangent to

\omega

and

\omega_A

F

and

F'

respectively. Furthermore, let

P

and

Q

be the intersections of

BI

with

EF

and

CI

with

EF

, respectively, and let

P'

and

Q'

be the intersections of

BI_A

with

E'F'

and

CI_A

with

E'F'

, respectively. Given that the circumradius of

\vartriangle ABC

is a, compute the maximum integer value of

BC

such that the area

[P QP'Q']

is less than or equal to

1

.
p21. Let

a

be the answer to Problem 19,

b

be the answer to Problem 20, and

c

be the answer to Problem 21.
Let

c

be a positive integer such that

gcd(b, c) = 1

. From each ordered pair

(x, y)

such that

x

and

y

are both integers, we draw two lines through that point in the

x-y

plane, one with slope

\frac{b}{c}

and one with slope

-\frac{c}{b}

. Given that the number of intersections of these lines in

[0, 1)^2

is a square number, what is the smallest possible value of

c

? Note that

[0, 1)^2

refers to all points

(x, y)

such that

0 \le x < 1

and

0 \le y < 1

Problem Statement