2012 AMC 12B Problems/Problem 25

Problem 25

Let $S=\{(x,y) : x\in \{0,1,2,3,4\}, y\in \{0,1,2,3,4,5\},\text{ and } (x,y)\ne (0,0)\}$ . 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 $\prod_{t\in T} f(t)?$

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{625}{144}\qquad\textbf{(C)}\ \frac{125}{24}\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ \frac{625}{24}$

Solution 1

Consider reflections. For any right triangle $ABC$ with the right labeling described in the problem, any reflection $A'B'C'$ labeled that way will give us $\tan CBA \cdot \tan C'B'A' = 1$ . First we consider the reflection about the line $y=2.5$ . Only those triangles $\subseteq T$ that have one vertex at $(0,5)$ do not reflect to a traingle $\subseteq T$ . Within those triangles, consider a reflection about the line $y=5-x$ . Then only those triangles $\subseteq T$ that have one vertex on the line $y=0$ do not reflect to a triangle $\subseteq T$ . So we only need to look at right triangles that have vertices $(0,5), (*,0), (*,*)$ . There are three cases:

Case 1: $A=(0,5)$ . Then $B=(*,0)$ is impossible.

Case 2: $B=(0,5)$ . Then we look for $A=(x,y)$ such that $\angle BAC=90^{\circ}$ and that $C=(*,0)$ . They are: $(A=(x,5), C=(x,0))$ , $(A=(3,2), C=(1,0))$ and $(A=(4,1), C=(3,0))$ . The product of their values of $\tan \angle CBA$ is $\frac{5}{1}\cdot \frac{5}{2} \cdot \frac{5}{3} \cdot \frac{5}{4} \cdot \frac{1}{4} \cdot \frac{2}{3} = \frac{625}{144}$ .

Case 3: $C=(0,5)$ . Then $A=(*,0)$ is impossible.

Therefore $\boxed{\textbf{(B)} \ \frac{625}{144}}$ is the answer.

Solution 2

This is just another way for the reasoning of solution 1. Picture the question to be a grid of unit squares instead of a coordinate system. (This is just to make visualization easier.) Define a "cell" to be a rectangle in the set of $S.$ For example, a cell can be $[asy] draw((0,0)--(0,1),black); draw((0,0)--(0,1),black); dot((0,0)); dot((3,7)); label("Produced with Asymptote "+version.VERSION,point(S),2S); [/asy]$

%\listfiles %\documentclass[11pt,a4paper]{article} \documentclass[11pt,border=2pt]{standalone} \usepackage[T1]{fontenc}

\usepackage{tikz} \usetikzlibrary{matrix}

\usepackage{lmodern} \usepackage{microtype}

\begin{document}

 \begin{tikzpicture}[
   line cap=round,
   line join=round,
 ]
   \matrix (grid) [
     matrix of nodes,
     column sep=-\pgflinewidth,
     row sep=-\pgflinewidth,
     nodes in empty cells,
     nodes={
       draw,
       anchor=center,
       minimum size=3em
     }
   ] {
     1 &  &  \\
       &  &  \\
       &  &  \\
   };
 \end{tikzpicture}

\end{document}

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2012amc12b/279

~dolphin7

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2012 AMC 12B Problems/Problem 25

Contents

Problem 25

Solution 1

Solution 2

Video Solution by Richard Rusczyk

See Also