Destroying houses
Tag(s):

## Easy-Medium, Expectation, Geometry, Probability

Problem
Editorial
Analytics

Hackerland has $n$ houses in the $xy$ plane . You are given $x$ and $y$ coordinates of all the $n$ houses. You plan to drop a bomb on hackerland. The bomb you possess has a destruction radius of $D$ . Formally, if you drop the bomb at point $(a,b)$, all houses within a distance $D$ from this point will be destroyed. You decide to drop the bomb at a point selected uniformly randomly inside a circle of radius $R$, centered at $(0,0)$. Find the expected value of the number of houses destroyed by the bomb.

Input
The first line contains $n$, $D$, and $R$ respectively.
It is followed by $n$ lines.The $i^{\text{th}}$ line contains two integers $x_i$, and $y_i$, representing $x$ and $y$ coordinates of the $i^{\text{th}}$ house.

Output
Output a single line containing the expected value of number of houses destroyed by the bomb to exactly 4 digits after the decimal point. This is equivalent to printf("%.4lf", value) in C.

Constraints
$n,D,R,x_i,y_i$ are integers
$1 \le n,D,R \le 10^6$
$-10^6 \le x_i,y_i \le 10^6$

SAMPLE INPUT
1 2 1
0 0

SAMPLE OUTPUT
1.0000

Explanation

Because $D = 2$, and $R=1$, no matter where you drop the bomb inside a circle of radius $R$ centered at $(0,0)$, the point $(0,0)$ will always be destroyed. So one house is destroyed no matter where the bomb is dropped. Hence, expected value of number of houses destroyed is $1$.

Time Limit: 3.0 sec(s) for each input file.
Memory Limit: 256 MB
Source Limit: 1024 KB
Marking Scheme: Marks are awarded when all the testcases pass.
Allowed Languages: C, C++, Clojure, C#, D, Erlang, F#, Go, Groovy, Haskell, Java, Java 8, JavaScript(Rhino), JavaScript(Node.js), Lisp, Lisp (SBCL), Lua, Objective-C, OCaml, Octave, Pascal, Perl, PHP, Python, Python 3, R(RScript), Racket, Ruby, Rust, Scala, Scala 2.11.8, Swift, Visual Basic

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## This Problem was Asked in

Challenge Name

August Circuits

OTHER PROBLEMS OF THIS CHALLENGE
• Math > Basic Math
• Math > Probablity
• Algorithms > Advanced Algorithms