MathDB

13. The triangle with vertices

(0,0), (a,b)

, and

(a,-b)

has area

10

. Find the sum of all possible positive integer values of

a

, given that

b

is a positive integer.
14. Elsa is venturing into the unknown. She stands on

(0,0)

in the coordinate plane, and each second, she moves to one of the four lattice points nearest her, chosen at random and with equal probability. If she ever moves to a lattice point she has stood on before, she has ventured back into the known, and thus stops venturing into the unknown from then on. After four seconds have passed, the probability that Elsa is still venturing into the unknown can be expressed as

\frac{a}{b}

in simplest terms. Find

a+b

.
(A lattice point is a point with integer coordinates.)
15. Let

ABCD

be a square with side length

4

. Points

X, Y,

and

Z

, distinct from points

A, B, C,

and

D

, are selected on sides

AD, AB,

and

CD

, respectively, such that

XY = 3, XZ = 4

, and

\angle YXZ = 90^{\circ}

. If

AX = \frac{a}{b}

in simplest terms, then find

a + b

Problem Statement