# Difference between revisions of "2012 AMC 12A Problems/Problem 23"

(Created page with "== Problem 23 == Let <math>S</math> be the square one of whose diagonals has endpoints <math>(0.1,0.7)</math> and <math>(-0.1,-0.7)</math>. A point <math>v=(x,y)</math> is chos...") |
(→Problem 23) |
||

Line 1: | Line 1: | ||

− | == Problem | + | == Problem == |

Let <math>S</math> be the square one of whose diagonals has endpoints <math>(0.1,0.7)</math> and <math>(-0.1,-0.7)</math>. A point <math>v=(x,y)</math> is chosen uniformly at random over all pairs of real numbers <math>x</math> and <math>y</math> such that <math>0 \le x \le 2012</math> and <math>0\le y\le 2012</math>. Let <math>T(v)</math> be a translated copy of <math>S</math> centered at <math>v</math>. What is the probability that the square region determined by <math>T(v)</math> contains exactly two points with integer coefficients in its interior? | Let <math>S</math> be the square one of whose diagonals has endpoints <math>(0.1,0.7)</math> and <math>(-0.1,-0.7)</math>. A point <math>v=(x,y)</math> is chosen uniformly at random over all pairs of real numbers <math>x</math> and <math>y</math> such that <math>0 \le x \le 2012</math> and <math>0\le y\le 2012</math>. Let <math>T(v)</math> be a translated copy of <math>S</math> centered at <math>v</math>. What is the probability that the square region determined by <math>T(v)</math> contains exactly two points with integer coefficients in its interior? | ||

<math> \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25 \qquad\textbf{(E)}\ 0.32 </math> | <math> \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25 \qquad\textbf{(E)}\ 0.32 </math> | ||

− | |||

== Solution == | == Solution == |

## Revision as of 18:29, 18 February 2012

## Problem

Let be the square one of whose diagonals has endpoints and . A point is chosen uniformly at random over all pairs of real numbers and such that and . Let be a translated copy of centered at . What is the probability that the square region determined by contains exactly two points with integer coefficients in its interior?