Combinatorics is a branch of mathematics chiefly concerned with counting discrete objects. For instance, how many ways can you pick two people out of a crowd of n people? Into how many regions can you divide a circular disk by connecting n points on its boundary with one another? How many cubes are in a pyramid with square layers ranging from 1 × 1 to n × n cubes?
TFigure 1:T If we connect six points on the boundary of a circle, at most 31 regions are created.
Many questions like these have answers that can be reduced to simple polynomials in n. The answer to the first question above is n(n-1)/2, or (n^2-n)/2. The answer to the second is (n^4-6n^3+23n^2-18n+24)/24. The answer to the third is n(n+1)(2n+1)/6, or (2n^3+3n^2+n)/6. We write these polynomials in a standard form, as a polynomial with integer coefficients divided by a positive integer denominator. These polynomials are answers to questions that can have integer answers only. But since they have fractional coefficients, they look as if they could produce non-integer results! Of course, evaluating these particular polynomials on a positive integer always results in an integer. For other polynomials of similar form, this is not necessarily true. It can be hard to tell the two cases apart. So that, naturally, is your task.
Input
The input consists of multiple test cases, each on a separate line. Each test case is an expression in the form (P)/D, where P is a polynomial with integer coefficients and D is a positive integer denominator. P is a sum of terms of the form Cn^E, where the coefficient C and the exponent E satisfy the following conditions:
1. E is an integer satisfying 0 ≤ E ≤ 100. If E is 0, then Cn^E is expressed as C. If E is 1, then Cn^E is expressed as Cn, unless C is 1 or -1. In those instances, Cn^E is expressed as n or -n.
2. C is an integer. If C is 1 or -1 and E is not 0 or 1, then the Cn^E will appear as n^E or -n^E.
3. Only non-negative C values that are not part of the first term in the polynomial are preceded by +.
4. Exponents in consecutive terms are strictly decreasing.
5. C and D fit in a 32-bit signed integer.
See the sample input for details.
Input is terminated by a line containing a single period.
OutputFor each test case, print the case number (starting with 1). Then print TAlways an integerT if the test casepolynomial evaluates to an integer for every positive integer n. Print TNot always an integerT otherwise. Print the output for separate test cases on separate lines. Your output should follow the same format as the sample output.
Sample Input
(n^2-n)/2
(2n^3+3n^2+n)/6
(-n^14-11n+1)/3
.
Output for the Sample Input
Case 1: Always an integer
Case 2: Always an integer
Case 3: Not always an integer
題目大概的意思是說：給定一個(gè)關(guān)于n的p次多項(xiàng)式，問該多項(xiàng)式是否為整值多項(xiàng)式。
根據(jù)定理：n次多項(xiàng)式f(n)是整值多項(xiàng)式當(dāng)且僅當(dāng)f(n)至少在n+1個(gè)連續(xù)的整數(shù)上都取整值。
只用將0-MAXPOW(取101）依次代入多項(xiàng)式的分子，并對(duì)分母d取模，判斷是否都為0即可。
至于為什么要取MAXPOW，而不是多項(xiàng)式f(n)的最大的次數(shù)max{pi}：為了使問題一般化，我們可以講所有的多項(xiàng)式都看成是MAXPOW次的，只不過當(dāng)次數(shù)p>max{pi}時(shí)，其對(duì)應(yīng)的系數(shù)ci全部為0，并不妨礙問題的解決。這樣一來，就不需要再額外求出f(n)的最大次數(shù)max{pi}，使程序得到簡化。
399645 |
2009-04-23 07:44:07 |
Accepted |
0.066 |
Minimum |
19193 |
C++ |
4119 - Always an integer |
1 #include <iostream>
2 using namespace std;
3
4 const int MAXPOW = 101;
5 int c[MAXPOW],d;
6 char ch;
7
8 int calculate(long long n){
9 int i;
10 long long ans=0;
11 for(i=MAXPOW;i>=0;i--)
12 ans=(ans*n+c[i])%d;
13 return (int)ans;
14 }
15 bool judge(){
16 int i;
17 for(i=0;i<=MAXPOW;i++)
18 if(calculate(i)) return false;
19 return true;
20 }
21 int main(){
22 int end,ca=1,sign,value,pow;
23 while(true){
24 ch=getchar();
25 if(ch=='.') break;
26 memset(c,0,sizeof(c));
27 while(true){
28 end=0,scanf(")%n",&end);
29 if(end) break;
30 scanf("+");
31 sign=0,value=1,scanf("-%n",&sign);
32 scanf("%d",&value);
33 if(sign) value=-value;
34 scanf("%nn%n^%n",&pow,&pow,&pow);
35 if(pow>1) scanf("%d",&pow);
36 c[pow]+=value;
37 }
38 scanf("/%d",&d);
39 getchar();
40 printf("Case %d: ",ca++);
41 puts(judge() ? "Always an integer" : "Not always an integer");
42 }
43 return 0;
44 }