正题

题目链接:https://www.luogu.org/problem/P5431

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题目大意

一个长度为$n$的序列$a$。

求$$(\sum _{i=1}^n\frac{k^i}{a_i})\%p$$

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解题思路

定义$$s_i=\prod _{i=1}^i{a}_i$$
$$\frac{1}{s_{i-1}}=\frac{1}{s_i}\ast a_i$$
$$\frac{1}{a_i}=\frac{1}{s_i}\ast s_{i-1}$$
然后我们可以$O(n)$推出$s_n$，然后求出$\frac{1}{s_n}$。然后倒推回去求出所有的$s_i$。

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$code$

#pragma GCC optimize(2)
%:pragma GCC optimize(3)
%:pragma GCC optimize("Ofast")
%:pragma GCC optimize("inline")
%:pragma GCC optimize("-fgcse")
%:pragma GCC optimize("-fgcse-lm")
%:pragma GCC optimize("-fipa-sra")
%:pragma GCC optimize("-ftree-pre")
%:pragma GCC optimize("-ftree-vrp")
%:pragma GCC optimize("-fpeephole2")
%:pragma GCC optimize("-ffast-math")
%:pragma GCC optimize("-fsched-spec")
%:pragma GCC optimize("unroll-loops")
%:pragma GCC optimize("-falign-jumps")
%:pragma GCC optimize("-falign-loops")
%:pragma GCC optimize("-falign-labels")
%:pragma GCC optimize("-fdevirtualize")
%:pragma GCC optimize("-fcaller-saves")
%:pragma GCC optimize("-fcrossjumping")
%:pragma GCC optimize("-fthread-jumps")
%:pragma GCC optimize("-funroll-loops")
%:pragma GCC optimize("-fwhole-program")
%:pragma GCC optimize("-freorder-blocks")
%:pragma GCC optimize("-fschedule-insns")
%:pragma GCC optimize("inline-functions")
%:pragma GCC optimize("-ftree-tail-merge")
%:pragma GCC optimize("-fschedule-insns2")
%:pragma GCC optimize("-fstrict-aliasing")
%:pragma GCC optimize("-fstrict-overflow")
%:pragma GCC optimize("-falign-functions")
%:pragma GCC optimize("-fcse-skip-blocks")
%:pragma GCC optimize("-fcse-follow-jumps")
%:pragma GCC optimize("-fsched-interblock")
%:pragma GCC optimize("-fpartial-inlining")
%:pragma GCC optimize("no-stack-protector")
%:pragma GCC optimize("-freorder-functions")
%:pragma GCC optimize("-findirect-inlining")
%:pragma GCC optimize("-fhoist-adjacent-loads")
%:pragma GCC optimize("-frerun-cse-after-loop")
%:pragma GCC optimize("inline-small-functions")
%:pragma GCC optimize("-finline-small-functions")
%:pragma GCC optimize("-ftree-switch-conversion")
%:pragma GCC optimize("-foptimize-sibling-calls")
%:pragma GCC optimize("-fexpensive-optimizations")
%:pragma GCC optimize("-funsafe-loop-optimizations")
%:pragma GCC optimize("inline-functions-called-once")
%:pragma GCC optimize("-fdelete-null-pointer-checks")
#include<cstdio>
#include<cstring>
#include<cctype>
#include<algorithm>
#define ll long long
using namespace std;
const ll N=5e6+10;
ll n,p,K,S,a[N],s[N],k[N],ans; 
__attribute__((optimize("O3"))) inline int read() {int x=0,f=1; char c=getchar();while(!isdigit(c)) {if(c=='-')f=-f;c=getchar();}while(isdigit(c)) x=(x<<1)+(x<<3)+c-48,c=getchar();return x*f;
}
ll power(ll x,ll b,ll p)
{ll ans=1;while(b){if(b&1) ans=ans*x%p;x=x*x%p;b>>=1;}return ans;
}
int main()
{n=read();p=read();K=read();S=1;k[0]=1;s[0]=1;for(int i=1;i<=n;i++){a[i]=read();S=S*a[i]%p;s[i]=S;k[i]=k[i-1]*K%p;}S=power(S,p-2,p);for(int i=n;i>=1;i--){ll inv=S*s[i-1]%p;S=S*a[i]%p;(ans+=k[i]*inv%p)%=p;}printf("%lld",ans);
}