正題

題目鏈接:https://www.luogu.org/problem/P5431

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題目大意

一個長度為$n$的序列$a$。

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

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解題思路

定義$$s_i=\prod _{i=1}^i{a}_i$$
$$\frac{1}{s_{i?1}}=\frac{1}{s_i}?a_i$$
$$\frac{1}{a_i}=\frac{1}{s_i}?s_{i?1}$$
然后我們可以$O(n)$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); }

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