codeforces #341 div2 C. Wet Shark and Flowers


C. Wet Shark and Flowers
time limit per test

2 seconds

memory limit per test

256 megabytes


standard input


standard output

There are n sharks who grow flowers for Wet Shark. They are all sitting around the table, such that sharks i and i + 1 are neighbours for all i from 1 to n - 1. Sharks n and 1 are neighbours too.

Each shark will grow some number of flowers si. For i-th shark value si is random integer equiprobably chosen in range from lito ri. Wet Shark has it’s favourite prime number p, and he really likes it! If for any pair of neighbouring sharks i and j the product si·sj is divisible by p, then Wet Shark becomes happy and gives 1000 dollars to each of these sharks.

At the end of the day sharks sum all the money Wet Shark granted to them. Find the expectation of this value.


The first line of the input contains two space-separated integers n and p (3 ≤ n ≤ 100 000, 2 ≤ p ≤ 109) — the number of sharks and Wet Shark’s favourite prime number. It is guaranteed that p is prime.

The i-th of the following n lines contains information about i-th shark — two space-separated integers li and ri(1 ≤ li ≤ ri ≤ 109), the range of flowers shark i can produce. Remember that si is chosen equiprobably among all integers fromli to ri, inclusive.


Print a single real number — the expected number of dollars that the sharks receive in total. You answer will be considered correct if its absolute or relative error does not exceed 10 - 6.

Namely: let’s assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if .

Sample test(s)





A prime number is a positive integer number that is divisible only by 1 and itself. 1 is not considered to be prime.

Consider the first sample. First shark grows some number of flowers from 1 to 2, second sharks grows from 420 to 421flowers and third from 420420 to 420421. There are eight cases for the quantities of flowers (s0, s1, s2) each shark grows:

  1. (1, 420, 420420): note that s0·s1 = 420, s1·s2 = 176576400, and s2·s0 = 420420. For each pair, 1000 dollars will be awarded to each shark. Therefore, each shark will be awarded 2000 dollars, for a total of 6000 dollars.
  2. (1, 420, 420421): now, the product s2·s0 is not divisible by 2. Therefore, sharks s0 and s2 will receive 1000 dollars, while shark s1 will receive 2000. The total is 4000.
  3. (1, 421, 420420): total is 4000
  4. (1, 421, 420421): total is 0.
  5. (2, 420, 420420): total is 6000.
  6. (2, 420, 420421): total is 6000.
  7. (2, 421, 420420): total is 6000.
  8. (2, 421, 420421): total is 4000.

The expected value is .

In the second sample, no combination of quantities will garner the sharks any money.




题意:有n个区间,第i个和第i+1个相邻,特别地,第n个和第1个也是相邻的。 每个区间选择每个数的可能性是相同的。如果一个组相邻区间选到的两个数的乘积可以被给定的素数p整除,那么这两个区间各得2000分,问所有区间的得分的期望总和是多少。


思路: 如果乘积是能被p整除,那么这两个数中至少有一个能被p整除。






我们不妨从初始划分。将区间[l,r]分成[1,l-1]和[1,r]  1到x中能被p整除的数的个数为x/p.

所以区间[l,r]中能被p整除的个数为 r/p-(l-1)/p;(考虑这种方法的一般意义,即如果要统计某个区间的某种数量,正面直接难统计的话,我们可以间接来统计,可能是从1,也可能是从0)