POJ 1840 Eqs

Posted 2022-04-06 zaq19970105

题目链接：https://vjudge.net/problem/POJ-1840

题目大意

　　有以下方程：

$$
\sum_i = 1^5 a_i * x_i^3 = 0, a_i \in [-50, 50], x_i \in [-50, 0) \cup (0, -50], a_i 和 x_i 都是整数
$$

　　现给定 5 个参数 a_i ，求方程有多少不同的解？

分析

　　直接枚举肯定超时，这时可以考虑移项，变成$\sum_i = 1^2 a_i * x_i^3 = \sum_i = 3^5 a_i * x_i^3$，这样全部枚举的复杂度就从 100⁴ 降到 100² + 100³了。

代码如下

  1 #include <cmath>
  2 #include <ctime>
  3 #include <iostream>
  4 #include <string>
  5 #include <vector>
  6 #include <cstdio>
  7 #include <cstdlib>
  8 #include <cstring>
  9 #include <queue>
 10 #include <map>
 11 #include <unordered_map>
 12 #include <set>
 13 #include <algorithm>
 14 #include <cctype>
 15 #include <stack>
 16 #include <deque>
 17 #include <list>
 18 #include <sstream>
 19 using namespace std;
 20  
 21 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
 22 #define Rep(i,n) for (int i = 0; i < (n); ++i)
 23 #define For(i,s,t) for (int i = (s); i <= (t); ++i)
 24 #define rFor(i,t,s) for (int i = (t); i >= (s); --i)
 25 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
 26 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
 27 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
 28 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)
 29  
 30 #define pr(x) cout << #x << " = " << x << "  "
 31 #define prln(x) cout << #x << " = " << x << endl
 32  
 33 #define LOWBIT(x) ((x)&(-x))
 34  
 35 #define ALL(x) x.begin(),x.end()
 36 #define INS(x) inserter(x,x.begin())
 37 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
 38 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // 删去 x 中所有 c 
 39 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
 40 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);
 41  
 42 #define ms0(a) memset(a,0,sizeof(a))
 43 #define msI(a) memset(a,inf,sizeof(a))
 44 #define msM(a) memset(a,-1,sizeof(a))
 45 
 46 #define MP make_pair
 47 #define PB push_back
 48 #define ft first
 49 #define sd second
 50  
 51 template<typename T1, typename T2>
 52 istream &operator>>(istream &in, pair<T1, T2> &p) 
 53     in >> p.first >> p.second;
 54     return in;
 55 
 56  
 57 template<typename T>
 58 istream &operator>>(istream &in, vector<T> &v) 
 59     for (auto &x: v)
 60         in >> x;
 61     return in;
 62 
 63  
 64 template<typename T1, typename T2>
 65 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) 
 66     out << "[" << p.first << ", " << p.second << "]" << "\n";
 67     return out;
 68 
 69 
 70 inline int gc()
 71     static const int BUF = 1e7;
 72     static char buf[BUF], *bg = buf + BUF, *ed = bg;
 73     
 74     if(bg == ed) fread(bg = buf, 1, BUF, stdin);
 75     return *bg++;
 76  
 77 
 78 inline int ri()
 79     int x = 0, f = 1, c = gc();
 80     for(; c<48||c>57; f = c==‘-‘?-1:f, c=gc());
 81     for(; c>47&&c<58; x = x*10 + c - 48, c=gc());
 82     return x*f;
 83 
 84 
 85 template<class T>
 86 inline string toString(T x) 
 87     ostringstream sout;
 88     sout << x;
 89     return sout.str();
 90 
 91 
 92 inline int toInt(string s) 
 93     int v;
 94     istringstream sin(s);
 95     sin >> v;
 96     return v;
 97 
 98 
 99 //min <= aim <= max
100 template<typename T>
101 inline bool BETWEEN(const T aim, const T min, const T max) 
102     return min <= aim && aim <= max;
103 
104  
105 typedef long long LL;
106 typedef unsigned long long uLL;
107 typedef pair< double, double > PDD;
108 typedef pair< int, int > PII;
109 typedef pair< int, PII > PIPII;
110 typedef pair< string, int > PSI;
111 typedef pair< int, PSI > PIPSI;
112 typedef set< int > SI;
113 typedef set< PII > SPII;
114 typedef vector< int > VI;
115 typedef vector< double > VD;
116 typedef vector< VI > VVI;
117 typedef vector< SI > VSI;
118 typedef vector< PII > VPII;
119 typedef map< int, int > MII;
120 typedef map< LL, int > MLLI;
121 typedef map< int, string > MIS;
122 typedef map< int, PII > MIPII;
123 typedef map< PII, int > MPIII;
124 typedef map< string, int > MSI;
125 typedef map< string, string > MSS;
126 typedef map< PII, string > MPIIS;
127 typedef map< PII, PII > MPIIPII;
128 typedef multimap< int, int > MMII;
129 typedef multimap< string, int > MMSI;
130 //typedef unordered_map< int, int > uMII;
131 typedef pair< LL, LL > PLL;
132 typedef vector< LL > VL;
133 typedef vector< VL > VVL;
134 typedef priority_queue< int > PQIMax;
135 typedef priority_queue< int, VI, greater< int > > PQIMin;
136 const double EPS = 1e-8;
137 const LL inf = 0x3fffffff;
138 const LL infLL = 0x3fffffffffffffffLL;
139 const LL mod = 1e9 + 7;
140 const int maxN = 1e5 + 7;
141 const LL ONE = 1;
142 const LL evenBits = 0xaaaaaaaaaaaaaaaa;
143 const LL oddBits = 0x5555555555555555;
144 
145 int a[6], ans; 
146 VI v3;
147 MII H;
148 
149 int main()
150     //freopen("MyOutput.txt","w",stdout);
151     //freopen("input.txt","r",stdin);
152     INIT();
153     For(i, 1, 5) cin >> a[i];
154     
155     For(i, -50, -1) v3.PB(i * i * i);
156     For(i, 1, 50) v3.PB(i * i * i);
157     
158     For(i, 0, 99) 
159         For(j, 0, 99)
160             ++H[-a[1] * v3[i] - a[2] * v3[j]];
161 
162     For(i, 0, 99) 
163         For(j, 0, 99) 
164             For(k, 0, 99) 
165                 int ret = a[3] * v3[i] + a[4] * v3[j] + a[5] * v3[k];
166                 if(H.find(ret) != H.end()) ans += H[ret];
167             
168     
169     cout << ans << endl;
170     return 0;
171 

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