Edit Problem

Problem Description Let \(S=\{(x,y) : x \in \{0,1,2,3,4\}, y \in \{0,1,2,3,4,5\}\), and \((x,y) \neq (0,0) \}\). Let \(T\) be the set of all right triangles whose vertices are in \(S\). For every right triangle \(t=\triangle ABC\) with vertices \(A\), \(B\), and \(C\) in counter-clockwise order and right angle at \(A\), let \(f(t)= \tan (\angle CBA)\). What is \[ \displaystyle \prod_{t \in T} f(t) \text{?} \] [asy] size((120)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((0,5)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((1,5)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((2,5)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((3,5)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); dot((4,5)); label("\(\circ\)", (0,0)); label("\(S\)", (-.7,2.5)); [/asy] $$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{625}{144} \qquad \textbf{(C)}\ \frac{125}{24} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{625}{24}$$

Diagram (TikZ Code)

Solution B

Tags

Difficulty

Source