結果
| 問題 | No.1501 酔歩 |
| ユーザー |
👑  tute7627
|
| 提出日時 | 2021-05-07 23:34:14 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 22,665 bytes |
| コンパイル時間 | 3,983 ms |
| コンパイル使用メモリ | 248,432 KB |
| 実行使用メモリ | 26,496 KB |
| 最終ジャッジ日時 | 2024-09-15 11:39:30 |
| 合計ジャッジ時間 | 7,887 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / judge1 |
（要ログイン）
テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | AC | 2 ms
10,624 KB |
| testcase_01 | AC | 2 ms
5,376 KB |
| testcase_02 | AC | 3 ms
5,376 KB |
| testcase_03 | AC | 3 ms
5,376 KB |
| testcase_04 | AC | 2 ms
5,376 KB |
| testcase_05 | TLE | - |
| testcase_06 | -- | - |
| testcase_07 | -- | - |
| testcase_08 | -- | - |
| testcase_09 | -- | - |
| testcase_10 | -- | - |
| testcase_11 | -- | - |
| testcase_12 | -- | - |
| testcase_13 | -- | - |
| testcase_14 | -- | - |
| testcase_15 | -- | - |
| testcase_16 | -- | - |
| testcase_17 | -- | - |
| testcase_18 | -- | - |
| testcase_19 | -- | - |
| testcase_20 | -- | - |
| testcase_21 | -- | - |
| testcase_22 | -- | - |
| testcase_23 | -- | - |
| testcase_24 | -- | - |
| testcase_25 | -- | - |
| testcase_26 | -- | - |
| testcase_27 | -- | - |
| testcase_28 | -- | - |
| testcase_29 | -- | - |
| testcase_30 | -- | - |
| testcase_31 | -- | - |
| testcase_32 | -- | - |
| testcase_33 | -- | - |
| testcase_34 | -- | - |
| testcase_35 | -- | - |
| testcase_36 | -- | - |
| testcase_37 | -- | - |
| testcase_38 | -- | - |
| testcase_39 | -- | - |
| testcase_40 | -- | - |
| testcase_41 | -- | - |
| testcase_42 | -- | - |
| testcase_43 | -- | - |
| testcase_44 | -- | - |
| testcase_45 | -- | - |
| testcase_46 | -- | - |
| testcase_47 | -- | - |
| testcase_48 | -- | - |
| testcase_49 | -- | - |
| testcase_50 | -- | - |
| testcase_51 | -- | - |
| testcase_52 | -- | - |
| testcase_53 | -- | - |
| testcase_54 | -- | - |
| testcase_55 | -- | - |
ソースコード
//#define _GLIBCXX_DEBUG
#include<bits/stdc++.h>
using namespace std;
#define endl '\n'
#define lfs cout<<fixed<<setprecision(10)
#define ALL(a) (a).begin(),(a).end()
#define ALLR(a) (a).rbegin(),(a).rend()
#define spa << " " <<
#define fi first
#define se second
#define MP make_pair
#define MT make_tuple
#define PB push_back
#define EB emplace_back
#define rep(i,n,m) for(ll i = (n); i < (ll)(m); i++)
#define rrep(i,n,m) for(ll i = (ll)(m) - 1; i >= (ll)(n); i--)
using ll = long long;
using ld = long double;
const ll MOD1 = 1e9+7;
const ll MOD9 = 998244353;
const ll INF = 1e18;
using P = pair<ll, ll>;
template<typename T> using PQ = priority_queue<T>;
template<typename T> using QP = priority_queue<T,vector<T>,greater<T>>;
template<typename T1, typename T2>bool chmin(T1 &a,T2 b){if(a>b){a=b;return true;}else return false;}
template<typename T1, typename T2>bool chmax(T1 &a,T2 b){if(a<b){a=b;return true;}else return false;}
ll median(ll a,ll b, ll c){return a+b+c-max({a,b,c})-min({a,b,c});}
void ans1(bool x){if(x) cout<<"Yes"<<endl;else cout<<"No"<<endl;}
void ans2(bool x){if(x) cout<<"YES"<<endl;else cout<<"NO"<<endl;}
void ans3(bool x){if(x) cout<<"Yay!"<<endl;else cout<<":("<<endl;}
template<typename T1,typename T2>void ans(bool x,T1 y,T2 z){if(x)cout<<y<<endl;else cout<<z<<endl;}
template<typename T1,typename T2,typename T3>void anss(T1 x,T2 y,T3 z){ans(x!=y,x,z);};
template<typename T>void debug(const T &v,ll h,ll w,string sv=" "){for(ll i=0;i<h;i++){cout<<v[i][0];for(ll j=1;j<w;j++)cout<<sv<<v[i][j];cout<<endl;}};
template<typename T>void debug(const T &v,ll n,string sv=" "){if(n!=0)cout<<v[0];for(ll i=1;i<n;i++)cout<<sv<<v[i];cout<<endl;};
template<typename T>void debug(const vector<T>&v){debug(v,v.size());}
template<typename T>void debug(const vector<vector<T>>&v){for(auto &vv:v)debug(vv,vv.size());}
template<typename T>void debug(stack<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(queue<T> st){while(!st.empty()){cout<<st.front()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(deque<T> st){while(!st.empty()){cout<<st.front()<<" ";st.pop_front();}cout<<endl;}
template<typename T>void debug(PQ<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(QP<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(const set<T>&v){for(auto z:v)cout<<z<<" ";cout<<endl;}
template<typename T>void debug(const multiset<T>&v){for(auto z:v)cout<<z<<" ";cout<<endl;}
template<typename T,typename V>void debug(const map<T,V>&v){for(auto z:v)cout<<"["<<z.first<<"]="<<z.second<<endl;}
template<typename T>vector<vector<T>>vec(ll x, ll y, T w){vector<vector<T>>v(x,vector<T>(y,w));return v;}
ll gcd(ll x,ll y){ll r;while(y!=0&&(r=x%y)!=0){x=y;y=r;}return y==0?x:y;}
vector<ll>dx={1,-1,0,0,1,1,-1,-1};vector<ll>dy={0,0,1,-1,1,-1,1,-1};
template<typename T>vector<T> make_v(size_t a,T b){return vector<T>(a,b);}
template<typename... Ts>auto make_v(size_t a,Ts... ts){return vector<decltype(make_v(ts...))>(a,make_v(ts...));}
template<typename T1, typename T2>ostream &operator<<(ostream &os, const pair<T1, T2>&p){return os << p.first << " " << p.second;}
template<typename T>ostream &operator<<(ostream &os, const vector<T> &v){for(auto &z:v)os << z << " ";cout<<"|"; return os;}
template<typename T>void rearrange(vector<int>&ord, vector<T>&v){
auto tmp = v;
for(int i=0;i<tmp.size();i++)v[i] = tmp[ord[i]];
}
template<typename Head, typename... Tail>void rearrange(vector<int>&ord,Head&& head, Tail&&... tail){
rearrange(ord, head);
rearrange(ord, tail...);
}
template<typename T> vector<int> ascend(const vector<T>&v){
vector<int>ord(v.size());iota(ord.begin(),ord.end(),0);
sort(ord.begin(),ord.end(),[&](int i,int j){return v[i]<v[j];});
return ord;
}
template<typename T> vector<int> descend(const vector<T>&v){
vector<int>ord(v.size());iota(ord.begin(),ord.end(),0);
sort(ord.begin(),ord.end(),[&](int i,int j){return v[i]>v[j];});
return ord;
}
ll FLOOR(ll n,ll div){return n>=0?n/div:(n-div+1)/div;}
ll CEIL(ll n,ll div){return n>=0?(n+div-1)/div:n/div;}
template<typename T>T min(const vector<T>&v){return *min_element(v.begin(),v.end());}
template<typename T>T max(const vector<T>&v){return *max_element(v.begin(),v.end());}
template<typename T>T acc(const vector<T>&v){return accumulate(v.begin(),v.end(),T(0));};
template<typename T>T reverse(const T &v){return T(v.rbegin(),v.rend());};
//mt19937 mt(chrono::steady_clock::now().time_since_epoch().count());
int popcount(ll x){return __builtin_popcountll(x);};
int poplow(ll x){return __builtin_ctzll(x);};
int pophigh(ll x){return 63 - __builtin_clzll(x);};
template<typename T>T poll(queue<T> &q){auto ret=q.front();q.pop();return ret;};
template<typename T>T poll(priority_queue<T> &q){auto ret=q.top();q.pop();return ret;};
template<typename T>T poll(QP<T> &q){auto ret=q.top();q.pop();return ret;};
template<typename T>T poll(stack<T> &s){auto ret=s.top();s.pop();return ret;};
template< typename T = int >
struct edge {
int to;
T cost;
int id;
edge():id(-1){};
edge(int to, T cost = 1, int id = -1):to(to), cost(cost), id(id){}
operator int() const { return to; }
};
template<typename T>
using Graph = vector<vector<edge<T>>>;
template<typename T>
Graph<T> readGraph(int n,int m,int indexed=1,bool directed=false,bool weighted=false){
Graph<T> ret(n);
for(int es = 0; es < m; es++){
int u,v,w=1;
cin>>u>>v;u-=indexed,v-=indexed;
if(weighted)cin>>w;
ret[u].emplace_back(v,w,es);
if(!directed)ret[v].emplace_back(u,w,es);
}
return ret;
}
template<typename T>
Graph<T> readParent(int n,int indexed=1,bool directed=true){
Graph<T>ret(n);
for(int i=1;i<n;i++){
int p;cin>>p;
p-=indexed;
ret[p].emplace_back(i);
if(!directed)ret[i].emplace_back(p);
}
return ret;
}
using cpx = complex<double>;
const double PI = acos(-1);
vector<cpx> roots = {{0, 0},
{1, 0}};
void ensure_capacity(int min_capacity) {
for (int len = roots.size(); len < min_capacity; len *= 2) {
for (int i = len >> 1; i < len; i++) {
roots.emplace_back(roots[i]);
double angle = 2 * PI * (2 * i + 1 - len) / (len * 2);
roots.emplace_back(cos(angle), sin(angle));
}
}
}
void fft(vector<cpx> &z, bool inverse) {
int n = z.size();
assert((n & (n - 1)) == 0);
ensure_capacity(n);
for (unsigned i = 1, j = 0; i < n; i++) {
int bit = n >> 1;
for (; j >= bit; bit >>= 1)
j -= bit;
j += bit;
if (i < j)
swap(z[i], z[j]);
}
for (int len = 1; len < n; len <<= 1) {
for (int i = 0; i < n; i += len * 2) {
for (int j = 0; j < len; j++) {
cpx root = inverse ? conj(roots[j + len]) : roots[j + len];
cpx u = z[i + j];
cpx v = z[i + j + len] * root;
z[i + j] = u + v;
z[i + j + len] = u - v;
}
}
}
if (inverse)
for (int i = 0; i < n; i++)
z[i] /= n;
}
vector<int> multiply_bigint(const vector<int> &a, const vector<int> &b, int base) {
int need = a.size() + b.size();
int n = 1;
while (n < need) n <<= 1;
vector<cpx> p(n);
for (size_t i = 0; i < n; i++) {
p[i] = cpx(i < a.size() ? a[i] : 0, i < b.size() ? b[i] : 0);
}
fft(p, false);
// a[w[k]] = (p[w[k]] + conj(p[w[n-k]])) / 2
// b[w[k]] = (p[w[k]] - conj(p[w[n-k]])) / (2*i)
vector<cpx> ab(n);
cpx r(0, -0.25);
for (int i = 0; i < n; i++) {
int j = (n - i) & (n - 1);
ab[i] = (p[i] * p[i] - conj(p[j] * p[j])) * r;
}
fft(ab, true);
vector<int> result(need);
long long carry = 0;
for (int i = 0; i < need; i++) {
long long d = (long long) (ab[i].real() + 0.5) + carry;
carry = d / base;
result[i] = d % base;
}
return result;
}
vector<int> multiply_mod(const vector<int> &a, const vector<int> &b, int m) {
int need = a.size() + b.size() - 1;
int n = 1;
while (n < need) n <<= 1;
vector<cpx> A(n);
for (size_t i = 0; i < a.size(); i++) {
int x = (a[i] % m + m) % m;
A[i] = cpx(x & ((1 << 15) - 1), x >> 15);
}
fft(A, false);
vector<cpx> B(n);
for (size_t i = 0; i < b.size(); i++) {
int x = (b[i] % m + m) % m;
B[i] = cpx(x & ((1 << 15) - 1), x >> 15);
}
fft(B, false);
vector<cpx> fa(n);
vector<cpx> fb(n);
for (int i = 0, j = 0; i < n; i++, j = n - i) {
cpx a1 = (A[i] + conj(A[j])) * cpx(0.5, 0);
cpx a2 = (A[i] - conj(A[j])) * cpx(0, -0.5);
cpx b1 = (B[i] + conj(B[j])) * cpx(0.5, 0);
cpx b2 = (B[i] - conj(B[j])) * cpx(0, -0.5);
fa[i] = a1 * b1 + a2 * b2 * cpx(0, 1);
fb[i] = a1 * b2 + a2 * b1;
}
fft(fa, true);
fft(fb, true);
vector<int> res(need);
for (int i = 0; i < need; i++) {
long long aa = (long long) (fa[i].real() + 0.5);
long long bb = (long long) (fb[i].real() + 0.5);
long long cc = (long long) (fa[i].imag() + 0.5);
res[i] = (aa % m + (bb % m << 15) + (cc % m << 30)) % m;
}
return res;
}
constexpr int digits(int base) noexcept {
return base <= 1 ? 0 : 1 + digits(base / 10);
}
constexpr int base = 1000'000'000;
constexpr int base_digits = digits(base);
constexpr int fft_base = 10'000; // fft_base^2 * n / fft_base_digits <= 10^15 for double
constexpr int fft_base_digits = digits(fft_base);
struct bigint {
// value == 0 is represented by empty z
vector<int> z; // digits
// sign == 1 <==> value >= 0
// sign == -1 <==> value < 0
int sign;
bigint(long long v = 0) {
*this = v;
}
bigint &operator=(long long v) {
sign = v < 0 ? -1 : 1;
v *= sign;
z.clear();
for (; v > 0; v = v / base)
z.push_back((int) (v % base));
return *this;
}
bigint(const string &s) {
read(s);
}
bigint &operator+=(const bigint &other) {
if (sign == other.sign) {
for (int i = 0, carry = 0; i < other.z.size() || carry; ++i) {
if (i == z.size())
z.push_back(0);
z[i] += carry + (i < other.z.size() ? other.z[i] : 0);
carry = z[i] >= base;
if (carry)
z[i] -= base;
}
} else if (other != 0 /* prevent infinite loop */) {
*this -= -other;
}
return *this;
}
friend bigint operator+(bigint a, const bigint &b) {
a += b;
return a;
}
bigint &operator-=(const bigint &other) {
if (sign == other.sign) {
if ((sign == 1 && *this >= other) || (sign == -1 && *this <= other)) {
for (int i = 0, carry = 0; i < other.z.size() || carry; ++i) {
z[i] -= carry + (i < other.z.size() ? other.z[i] : 0);
carry = z[i] < 0;
if (carry)
z[i] += base;
}
trim();
} else {
*this = other - *this;
this->sign = -this->sign;
}
} else {
*this += -other;
}
return *this;
}
friend bigint operator-(bigint a, const bigint &b) {
a -= b;
return a;
}
bigint &operator*=(int v) {
if (v < 0)
sign = -sign, v = -v;
for (int i = 0, carry = 0; i < z.size() || carry; ++i) {
if (i == z.size())
z.push_back(0);
long long cur = (long long) z[i] * v + carry;
carry = (int) (cur / base);
z[i] = (int) (cur % base);
}
trim();
return *this;
}
bigint operator*(int v) const {
return bigint(*this) *= v;
}
friend pair<bigint, bigint> divmod(const bigint &a1, const bigint &b1) {
int norm = base / (b1.z.back() + 1);
bigint a = a1.abs() * norm;
bigint b = b1.abs() * norm;
bigint q, r;
q.z.resize(a.z.size());
for (int i = (int) a.z.size() - 1; i >= 0; i--) {
r *= base;
r += a.z[i];
int s1 = b.z.size() < r.z.size() ? r.z[b.z.size()] : 0;
int s2 = b.z.size() - 1 < r.z.size() ? r.z[b.z.size() - 1] : 0;
int d = (int) (((long long) s1 * base + s2) / b.z.back());
r -= b * d;
while (r < 0)
r += b, --d;
q.z[i] = d;
}
q.sign = a1.sign * b1.sign;
r.sign = a1.sign;
q.trim();
r.trim();
return {q, r / norm};
}
friend bigint sqrt(const bigint &a1) {
bigint a = a1;
while (a.z.empty() || a.z.size() % 2 == 1)
a.z.push_back(0);
int n = a.z.size();
int firstDigit = (int) ::sqrt((double) a.z[n - 1] * base + a.z[n - 2]);
int norm = base / (firstDigit + 1);
a *= norm;
a *= norm;
while (a.z.empty() || a.z.size() % 2 == 1)
a.z.push_back(0);
bigint r = (long long) a.z[n - 1] * base + a.z[n - 2];
firstDigit = (int) ::sqrt((double) a.z[n - 1] * base + a.z[n - 2]);
int q = firstDigit;
bigint res;
for (int j = n / 2 - 1; j >= 0; j--) {
for (;; --q) {
bigint r1 = (r - (res * 2 * base + q) * q) * base * base +
(j > 0 ? (long long) a.z[2 * j - 1] * base + a.z[2 * j - 2] : 0);
if (r1 >= 0) {
r = r1;
break;
}
}
res *= base;
res += q;
if (j > 0) {
int d1 = res.z.size() + 2 < r.z.size() ? r.z[res.z.size() + 2] : 0;
int d2 = res.z.size() + 1 < r.z.size() ? r.z[res.z.size() + 1] : 0;
int d3 = res.z.size() < r.z.size() ? r.z[res.z.size()] : 0;
q = (int) (((long long) d1 * base * base + (long long) d2 * base + d3) / (firstDigit * 2));
}
}
res.trim();
return res / norm;
}
bigint operator/(const bigint &v) const {
return divmod(*this, v).first;
}
bigint operator%(const bigint &v) const {
return divmod(*this, v).second;
}
bigint &operator/=(int v) {
if (v < 0)
sign = -sign, v = -v;
for (int i = (int) z.size() - 1, rem = 0; i >= 0; --i) {
long long cur = z[i] + rem * (long long) base;
z[i] = (int) (cur / v);
rem = (int) (cur % v);
}
trim();
return *this;
}
bigint operator/(int v) const {
return bigint(*this) /= v;
}
int operator%(int v) const {
if (v < 0)
v = -v;
int m = 0;
for (int i = (int) z.size() - 1; i >= 0; --i)
m = (int) ((z[i] + m * (long long) base) % v);
return m * sign;
}
bigint &operator*=(const bigint &v) {
*this = *this * v;
return *this;
}
bigint &operator/=(const bigint &v) {
*this = *this / v;
return *this;
}
bigint &operator%=(const bigint &v) {
*this = *this % v;
return *this;
}
bool operator<(const bigint &v) const {
if (sign != v.sign)
return sign < v.sign;
if (z.size() != v.z.size())
return z.size() * sign < v.z.size() * v.sign;
for (int i = (int) z.size() - 1; i >= 0; i--)
if (z[i] != v.z[i])
return z[i] * sign < v.z[i] * sign;
return false;
}
bool operator>(const bigint &v) const {
return v < *this;
}
bool operator<=(const bigint &v) const {
return !(v < *this);
}
bool operator>=(const bigint &v) const {
return !(*this < v);
}
bool operator==(const bigint &v) const {
return !(*this < v) && !(v < *this);
}
bool operator!=(const bigint &v) const {
return *this < v || v < *this;
}
void trim() {
while (!z.empty() && z.back() == 0)
z.pop_back();
if (z.empty())
sign = 1;
}
bool isZero() const {
return z.empty();
}
friend bigint operator-(bigint v) {
if (!v.z.empty())
v.sign = -v.sign;
return v;
}
bigint abs() const {
return sign == 1 ? *this : -*this;
}
long long longValue() const {
long long res = 0;
for (int i = (int) z.size() - 1; i >= 0; i--)
res = res * base + z[i];
return res * sign;
}
friend bigint gcd(const bigint &a, const bigint &b) {
return b.isZero() ? a : gcd(b, a % b);
}
friend bigint lcm(const bigint &a, const bigint &b) {
return a / gcd(a, b) * b;
}
void read(const string &s) {
sign = 1;
z.clear();
int pos = 0;
while (pos < s.size() && (s[pos] == '-' || s[pos] == '+')) {
if (s[pos] == '-')
sign = -sign;
++pos;
}
for (int i = (int) s.size() - 1; i >= pos; i -= base_digits) {
int x = 0;
for (int j = max(pos, i - base_digits + 1); j <= i; j++)
x = x * 10 + s[j] - '0';
z.push_back(x);
}
trim();
}
friend istream &operator>>(istream &stream, bigint &v) {
string s;
stream >> s;
v.read(s);
return stream;
}
friend ostream &operator<<(ostream &stream, const bigint &v) {
if (v.sign == -1)
stream << '-';
stream << (v.z.empty() ? 0 : v.z.back());
for (int i = (int) v.z.size() - 2; i >= 0; --i)
stream << setw(base_digits) << setfill('0') << v.z[i];
return stream;
}
static vector<int> convert_base(const vector<int> &a, int old_digits, int new_digits) {
vector<long long> p(max(old_digits, new_digits) + 1);
p[0] = 1;
for (int i = 1; i < p.size(); i++)
p[i] = p[i - 1] * 10;
vector<int> res;
long long cur = 0;
int cur_digits = 0;
for (int v : a) {
cur += v * p[cur_digits];
cur_digits += old_digits;
while (cur_digits >= new_digits) {
res.push_back(int(cur % p[new_digits]));
cur /= p[new_digits];
cur_digits -= new_digits;
}
}
res.push_back((int) cur);
while (!res.empty() && res.back() == 0)
res.pop_back();
return res;
}
bigint operator*(const bigint &v) const {
if (min(z.size(), v.z.size()) < 150)
return mul_simple(v);
bigint res;
res.sign = sign * v.sign;
res.z = multiply_bigint(convert_base(z, base_digits, fft_base_digits),
convert_base(v.z, base_digits, fft_base_digits), fft_base);
res.z = convert_base(res.z, fft_base_digits, base_digits);
res.trim();
return res;
}
bigint mul_simple(const bigint &v) const {
bigint res;
res.sign = sign * v.sign;
res.z.resize(z.size() + v.z.size());
for (int i = 0; i < z.size(); ++i)
if (z[i])
for (int j = 0, carry = 0; j < v.z.size() || carry; ++j) {
long long cur = res.z[i + j] + (long long) z[i] * (j < v.z.size() ? v.z[j] : 0) + carry;
carry = (int) (cur / base);
res.z[i + j] = (int) (cur % base);
}
res.trim();
return res;
}
};
mt19937 rng(1);
bigint random_bigint(int n) {
string s;
for (int i = 0; i < n; i++) {
s += uniform_int_distribution<int>('0', '9')(rng);
}
return bigint(s);
}
struct RationalNumber{
using I=bigint;
I a, b;
RationalNumber(){
a = 0,b = 1;
}
RationalNumber(I p, I q):a(p), b(q){// a / b
reduction();
};
RationalNumber(I p):a(p), b(1){// a / b
};
void reduction(){
I g = gcd(a, b);
a /= g, b /= g;
if(b < 0)a *= -1,b *= -1;
}
RationalNumber &operator+=(const RationalNumber &x){
I q = b / gcd(b, x.b) * x.b;
a = a * (q / b) + x.a * (q / x.b);
b = q;
reduction();
return *this;
}
RationalNumber &operator-=(const RationalNumber &x){
I q = b / gcd(b, x.b) * x.b;
a = a * (q / b) - x.a * (q / x.b);
b = q;
reduction();
return *this;
}
RationalNumber &operator*=(const RationalNumber &x){
I q1 = gcd(a, x.b), q2 = gcd(b, x.a);
a = (a / q1) * (x.a / q2), b = (b / q2) * (x.b / q1);
return *this;
}
RationalNumber &operator/=(const RationalNumber &x){
I q1 = gcd(a, x.a), q2 = gcd(b, x.b);
a = (a / q1) * (x.b / q2), b = (b / q2) * (x.a / q1);
return *this;
}
RationalNumber operator+(const RationalNumber &x)const{
return RationalNumber(*this) += x;
}
RationalNumber operator-(const RationalNumber &x)const{
return RationalNumber(*this) -= x;
}
RationalNumber operator*(const RationalNumber &x)const{
return RationalNumber(*this) *= x;
}
RationalNumber operator/(const RationalNumber &x)const{
return RationalNumber(*this) /= x;
}
RationalNumber operator-()const{
return RationalNumber(-a, b);
}
bool operator==(const RationalNumber &x)const{
return a * x.b == b * x.a;
};
bool operator!=(const RationalNumber &x)const{
return !(*this == x);
};
bool operator<(const RationalNumber &x)const{
return a * x.b < b * x.a;
}
bool operator>(const RationalNumber &x)const{
return a * x.b > b * x.a;
}
bool operator<=(const RationalNumber &x)const{
return *this < x || *this == x;
}
bool operator>=(const RationalNumber &x)const{
return *this > x || *this == x;
}
friend ostream &operator<<(ostream &os, RationalNumber x){
return os << x.a << "/" << x.b;
}
};
using rat=RationalNumber;
int main(){
cin.tie(nullptr);
ios_base::sync_with_stdio(false);
ll res=0,buf=0;
bool judge = true;
ll n,k;cin>>n>>k;k--;
vector<ll>a(n);
rep(i,0,n)cin>>a[i];
vector<rat>b(n),c(n);
b[0]=bigint(1),c[0]=bigint(0);
rep(i,1,n-1){
rat p(a[i-1],a[i+1]+a[i-1]);
rat q(a[i+1],a[i+1]+a[i-1]);
b[i]=p*b[i-1]/(rat(bigint(1))-p*c[i-1]);
c[i]=q/(rat(bigint(1))-p*c[i-1]);
}
vector<rat>d(n);
d[n-1]=bigint(0);
rrep(i,0,n-1){
d[i]=b[i]+c[i]*d[i+1];
}
if(d[k]==rat(bigint(1)))cout<<0<<endl;
else cout<<rat(bigint(1))-d[k]<<endl;
return 0;
}