Grodno 2015 (Urozero May 2015 Day 5)
A. Palindromes
留坑。
B. Modules
将$a$排序，那么最优解中$a_n$一定放在前后一个，且前面$n-1$个每次要么放最小的，要么放最大的，区间DP即可。
#include<cstdio>
#include<algorithm>
#include<bitset>
#include<cmath>
using namespace std;
const int N=505,M=1000;
int n,i,j,k,a[N],ans;
bitset<M>f[N][N];
int main(){
scanf("%d",&n);
if(n<1||n>300)while(1);
for(i=1;i<=n;i++){
scanf("%d",&a[i]);
if(a[i]<1||a[i]>300)while(1);
}
sort(a+1,a+n+1);
if(n==1)return printf("%d",a[1]),0;
if(n==2)return printf("%d",abs(a[1]-a[2])),0;
n--;
f[1][n+1][a[1]]=1;
f[0][n][a[n]]=1;
for(i=0;i<=n+1;i++)for(j=n+1;j>i;j--){
if(i+1<j)for(k=0;k<M;k++){
if(f[i][j][k]){
if(abs(k-a[i+1])>=M)while(1);
if(abs(k-a[j-1])>=M)while(1);
f[i+1][j][abs(k-a[i+1])]=1;
f[i][j-1][abs(k-a[j-1])]=1;
}
}
if(i+1==j)for(k=0;k<M;k++)if(f[i][j][k])ans=max(ans,abs(k-a[n+1]));
}
/*ans=abs(a[n-1]-a[n-2]);
for(i=n-3;i;i--)ans=abs(ans-a[i]);
ans=abs(ans-a[n]);*/
printf("%d",ans);
}
/*
8
4 7 3 1 4 6 10 5
*/
C. Rooks
根据题意直接计算。
#include<stdio.h>
#include<iostream>
#include<string.h>
#include<string>
#include<ctype.h>
#include<math.h>
#include<set>
#include<map>
#include<vector>
#include<queue>
#include<bitset>
#include<algorithm>
#include<time.h>
using namespace std;
void fre() { }
#define MS(x, y) memset(x, y, sizeof(x))
#define ls o<<1
#define rs o<<1|1
typedef long long LL;
typedef unsigned long long UL;
typedef unsigned int UI;
template <class T1, class T2>inline void gmax(T1 &a, T2 b) { if (b > a)a = b; }
template <class T1, class T2>inline void gmin(T1 &a, T2 b) { if (b < a)a = b; }
const int N = 1e5 + 10, M = 0, Z = 1e9 + 7, inf = 0x3f3f3f3f;
template <class T1, class T2>inline void gadd(T1 &a, T2 b) { a = (a + b) % Z; }
int casenum, casei;
int n; int main()
{
while(~scanf("%d", &n)){
printf("%d\n", (1 + n * n) * n * n / 2 - (1 + n * n) * n / 2);
} return 0;
}
/*
【trick&&吐槽】 【题意】 【分析】 【时间复杂度&&优化】 */
D. Triangles
发现$C$是完全平方数，枚举$C$后容斥统计区间内与其互质的数字个数即可。
#include<stdio.h>
#include<iostream>
#include<string.h>
#include<string>
#include<ctype.h>
#include<math.h>
#include<set>
#include<map>
#include<vector>
#include<queue>
#include<bitset>
#include<algorithm>
#include<time.h>
using namespace std;
void fre() { }
#define MS(x, y) memset(x, y, sizeof(x))
#define ls o<<1
#define rs o<<1|1
typedef long long LL;
typedef unsigned long long UL;
typedef unsigned int UI;
template <class T1, class T2>inline void gmax(T1 &a, T2 b) { if (b > a)a = b; }
template <class T1, class T2>inline void gmin(T1 &a, T2 b) { if (b < a)a = b; }
const int N = 1e5 + 10, M = 0, Z = 1e9 + 7, inf = 0x3f3f3f3f;
template <class T1, class T2>inline void gadd(T1 &a, T2 b) { a = (a + b) % Z; }
int casenum, casei;
int P;
bool ok(LL a, LL c)
{
LL b = (a * a - c * c) / c;
return a + b + c <= P;
}
int v[N],p[N],tot,mu[N];
vector<int>mud[N];
inline int askcoprime(int x,int l,int r){
if(l>r)return 0;
//x>=1,l>=1
int ret=0;
for(int i=0;i<mud[x].size();i++){
int d=mud[x][i];
ret+=mu[d]*(r/d-(l-1)/d);
}
return ret;
}
void sieve(){
int i,j;
for(mu[1]=1,i=2;i<N;i++){
if(!v[i])mu[i]=-1,p[tot++]=i;
for(j=0;j<tot&&1LL*i*p[j]<N;j++){
v[i*p[j]]=1;
if(i%p[j])mu[i*p[j]]=-mu[i];else break;
}
}
for(i=1;i<N;i++)if(mu[i])for(j=i;j<N;j+=i)mud[j].push_back(i);
}
//
int main()
{
sieve();
while(~scanf("%d", &P))
{
LL ans = 0;
//x = sqrt(c)
for(LL x = 1; x * x <= P; ++x)
{
LL c = x * x; LL L = x + 1;
LL R = x; LL l = x + 1;
LL r = x + x - 1;
while(l <= r)
{
LL mid = (l + r >> 1);
if(ok(mid * x, c))
{
R = mid;
l = mid + 1;
}
else
{
r = mid - 1;
}
}
//a is [L, R] if(L <= R)
{
ans += askcoprime(x, L, R);
}
}
printf("%lld\n", ans % Z);
}
return 0;
}
/*
【trick&&吐槽】 【题意】 【分析】 【时间复杂度&&优化】 */
E. Supercomputer
模拟竖式乘法构造方案即可。
F. Simple problem
将$P$分解质因数，对于每个质数判断是否有解。注意$K$个数允许相同，因此随便找一个解然后复制$K$份即可。
#include<cstdio>
#include<iostream>
#include<queue>
#include<set>
#include<cstdlib>
using namespace std;
typedef long long ll;
ll K,A,B,P;set<ll>T;
void OK(){
puts("YES");
for(set<ll>::iterator it=T.begin();it!=T.end();it++)printf("%lld ",*it);
exit(0);
}
inline void gao(ll x){
ll l=A/x*x;
while(1){
if(l>=B)return;
if(l>A&&l<B){
puts("YES");
for(ll i=1;i<=K;i++)printf("%lld ",l);
exit(0);
T.insert(l);
//if(T.size()==K)OK();
}
l+=x;
}
}
int main(){
cin>>A>>B>>K>>P;
if(P==1)return puts("NO"),0;
for(ll i=2;i*i<=P;i++)if(P%i==0){
gao(i);
while(P%i==0)P/=i;
}
if(P>1)gao(P);
puts("NO");
}
G. Ancient problem
解同余方程即可得到指数。
#include<cstdio>
typedef long long LL;
const int N=1010, Z = 1e9 + 7;
int n;
LL p[N], a[N], ans; LL extend_gcd(LL a, LL b, LL &x, LL &y)
{
if(b == 0){
x = 1; y = 0;
return a;
}
else{
LL r = extend_gcd(b, a % b, y, x);
y -= x * (a / b);
return r;
}
} LL line_mod_equation(LL a, LL b, LL n){
LL x, y;
LL d = extend_gcd(a, n, x, y);
LL ans;
if(b % d == 0){
x %= n; x += n; x %= n;
ans = x * (b / d) % (n / d);
}
return ans;
} LL quickmul(LL x, LL y)
{
LL ans = 1;
while(y){
if(y & 1) ans = ans * x % Z;
y >>= 1;
x = x * x % Z;
}
return ans;
} int main(){
while(~ scanf("%d", &n)){
for(int i = 1; i <= n; i ++){
scanf("%lld", &p[i]);
}
ans = 1;
for(int i = 1; i <= n; i ++){
a[i] = 1;
for(int j = 1; j <= n; j ++){
if(i != j) a[i] = a[i] * p[j] % p[i];
}
LL y = line_mod_equation(a[i], 1, p[i]);
a[i] = 1;
for(int j = 1; j <= n; j ++){
if(i != j) a[i] = a[i] * p[j] % (Z - 1);
}
ans = ans * quickmul(p[i], a[i] * y % (Z - 1)) % Z;
}
printf("%lld\n", ans);
}
return 0;
}
H. Combinatorics
答案为总方案数减去不合法的，总方案数显然是组合数，而不合法的可以看成从大到小依次插入序列，也可以由$O(n)$次组合数计算。
对于模$10^9+7$的组合数，可以使用Lucas定理配合分块打表计算。
#include<cstdio>
typedef long long ll;
const int M=300000,P=1000000007;
int n,i,all,sub;ll a[55],s[55];
int f[]={1,373281933,679209364,730342129,363599536,448516387,663484182,229611066,585806050,816761238,5832229,932799119,150478174,358784546,544918953,59433257,278955367,293296655,473966292,214054614,316220877,652382508,405577532,257438891,24521750,304661605,14422365,808427728,220087090,295620142,980250487,839684660,764421236,345911460,152611299,681184885,140864216,733842728,461157687,642502601,95936601,382610562,569582206,784835785,959173997,391177708,508362944,33592533,275742306,889166166,598816162,344176836,564190300,977909564,632048012,810640932,851161246,551739578,605016577,566161015,626497524,512332467,475501650,652967545,52298443,406755567,125625755,708750650,385457627,899820643,245341950,725000295,754437203,970854495,524308015,913230882,792180189,192185054,788944266,734897886,968999,642903797,511710008,3525929,979137619,328019447,30153596,610696318,353049206,500956856,881746146,87892573,919203761,122316469,910061172,560607302,542955556,31318378,496532173,778006566,76479948,126712923,941723214,85186984,685075984,30473806,581968000,348586163,451116854,448457825,886294336,570443478,695702195,107560517,229827108,771153434,296974594,546178363,204857297,656049509,544075857,876081033,632593345,642498983,854821744,468116664,466541843,644759890,198998756,836550874,82926604,848688381,924258126,960862718,535478979,320608814,63932312,571781438,305569485,710807771,721996174,979809664,526281212,44756165,522961673,814440466,589831644,21771073,268164416,364347439,172827403,106892865,169338716,455503344,675113999,983975503,475501867,852493922,415376,146628257,294587621,103228118,875018827,274155554,172340144,676243000,43217172,912590546,16783532,95325827,497478507,511176934,581794056,638088240,404666335,726686688,850677980,412956409,12568481,998406941,228338256,703470628,52378412,21373798,445935909,583676448,650765251,957424115,808076198,783200314,746536789,996955070,152366450,132350382,415355660,249780014,333236625,504223583,570866962,938204550,27368307,777338999,849148138,484723081,422227498,44635470,243833737,437862846,89372661,152019116,5974669,627402642,833342486,505819550,468608739,679975009,59143878,313702365,635032673,311539953,86609485,945754267,586359929,631326068,724008021,607649469,135711291,996881960,324648749,639334305,371263376,745897488,61732363,75142816,279914778,527225600,9409022,896155508,358050946,713802282,808412394,703420540,724209709,768955866,79329551,409420559,930161980,776363214,62785084,275062583,1669644,863596908,929666976,228137471,974936549,470618701,703087658,845285046,880634238,858493510,595143852,727281230,666743467,760927536,243910108,727223125,469594245,436778250,274161939,161004891,526807168,320466978,54080847,249481023,754963083,775147177,588618211,982508606,293401592,795558854,459188207,458068555,304228537,817509706,947251800,669251006,701643375,522396951,349343101,413937464,840398449,445438821,704289349,405638730,121102289,556957390,961729079,460879693,270409779,491891996,888050723,2840333,756462483,552487230,247544137,196710441,489872437,289842964,534030848,427641633,440902696,822593212,522303856,322754828,288079694,807209645,111668651,116180433,805525713,369444386,56305184,538549429,855064933,408481567,706513987,270603374,888874515,869737690,450119928,679238566,804805577,651180933,546799298,179149268,870230497,157803266,83687737,949485734,660162137,196609367,934814019,759805090,400707264,223692929,30684171,30657162,122464431,98740827,888156482,932064720,112249297,746200907,324630241,589132443,726576875,954976637,987095152,437453830,984659637,754518701,47401357,826027689,112861607,696470764,497956225,250361638,40151094,76721110,414608027,530157868,60168088,241724280,516804319,789824550,376501201,261319829,79852813,845029536,168948455,542422449,14235602,377668907,218851785,212574473,191651583,174624465,784685853,627701264,212661821,308722031,697066277,429594588,721056741,778453854,560478118,228266417,859346814,789370638,258032745,557936424,661224977,845620414,610644835,612842107,925041977,356438399,376285455,147688544,731950688,115172504,802214249,935517277,310280612,216317674,6811130,877125482,500440019,574835154,124878394,550028962,583967898,723291751,567003562,225111059,89921218,672118586,873138566,684564333,787539126,977891023,972424306,803353065,682495521,30664649,834892236,362493992,714130414,173532562,465697749,141265712,243276029,877786708,716734025,378511070,787117889,437447268,907685793,459332010,731397402,228400287,797848181,379570268,515174333,713434896,455781723,435508211,935521357,128010060,447063192,111481282,159554990,705858408,802837814,186918049,14211262,449205554,577063899,331605728,40607687,903447414,250388809,85199112,727934933,19819402,737738750,748359168,585552623,705998863,296457528,292850529,82104855,764232222,761748690,747471725,190349508,132581614,862776555,424920050,803726808,342432253,244109365,478743321,155570288,277342089,397066469,193109503,177573201,23990272,793806704,172909004,261384175,866948624,757717391,159233180,439520189,600281990,64851034,38555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86433112,818254617,198532616,870605651,646184883,327890704,438569183,364380585,624902060,63375050,381703446,897560127,903288230,542351792,143745971,907002124,502614804,260466949,729521334,827102807,34223419,880521254,281744795,469449264,52363853,52097767,585145563,237120480,905415848,680015907,969123061,964404181,563045796,333219547,789556231,638406068,918664302,213092254,740938590,381785483,966640179,387628955,662565976,866444111,699722694,508943810,792638611,793307102,970104063,826766689,506416432,3624695,443183564,312329365,907388810,901161015,948724148,362928234,380480024,289805123,767370785,62659548,801639277,609048943,26804576,989024750,274007936,778983779,814306574,315177422,811826264,991853918,323582350,973313674,759005373,82441236,892050437,698308420,678383330,492881183,221350454,808369065,801413581,500512522,976705957,412675227,349322665,713258160,776502975,803331189,772391340,445933418,131731042,660234457,486620721,3991476,878640901,804861409,322961008,364058973,506421625,354103106,260831555,359056292,12618703,344608141,50477806,217298111,524150252,395850518,235142930,19270208,366761197,187697925,120685561,258450659,446634274,954913,154622141,186631946,33495801,181835443,338471174,808138529,285892973,437310030,371007759,10430738,652336595,96008232,34847604,826784106,864696066,304794421,918564215,282183879,10393336,567473423,326708726,832860641,817925464,816547758,426786760,636346220,217118148,790758484,89856602,946149627,420034479,792270972,600743238,514349511,71050762,104677203,860090592,205004338,630391823,462880311,793657339,4283891,598513367,458983926,411673788,354893353,712921767,716425891,842530849,965785236,311669452,108224705,560260153,633400565,545638190,790523771,318580210,668677123,910026008,881482632,42587322,437155518,792299597,36670825,84428155,929269243,944491355,528255018,629894526,512634493,445396258,186968777,320428001,924265341,860842034,296589615,520374634,171989322,457294725,967284733,679659893,63126637,603739686,812501999,671517513,916660987,858814345,246311854,164992964,165393390,8195350,483975089,509406823,114644779,685723887,53102426,617666448,207172258,623151847,532702135,768941501,429707160,198803488,779734593,642939667,53635336,960584207,116164597,197001309,536841269,368324673,630057303,792914276,386541987,994065976,627500870,573443769,599685896,966533706,301196854,759137421,371930700,775556523,767790141,20211256,379943099,792894374,824619763,946198358,373451745,596952378,752243809,114955307,925139828,268761837,860551136,818476649,578471809,348246100,576368335,52979524,120815665,592296500,119895268,98016105,291993453,23895805,206526983,441442575,847549272,724553294,374116154,782816485,550507082,420567331,976517503,555103342,946033733,886354425,216046746,283129306,85104325,312720425,877002932,383521365,765080455,599755664,894658473,474857680,780882948,311490118,74434897,433350723,35784547,128900492,973737505,168759145,139066955,353658242,869544707,778368564,590393084,926316480};
inline int pow(int a,int b){
int t=1;
for(;b;b>>=1,a=1LL*a*a%P)if(b&1)t=1LL*t*a%P;
return t;
}
inline int F(int n){
int ret=f[n/M];
for(int i=n/M*M+1;i<=n;i++)ret=1LL*ret*i%P;
return ret;
}
inline ll C(ll n,ll m){
if(n<m)return 0;
if(n==m)return 1;
if(!m)return 1;
return 1LL*F(n)*pow(F(m),P-2)%P*pow(F(n-m),P-2)%P;
}
int lucas(ll n,ll m){
if(n<m)return 0;
if(n==m)return 1;
if(!m)return 1;
return 1LL*C(n%P,m%P)*lucas(n/P,m/P)%P;
}
int main(){
scanf("%d",&n);
for(i=1;i<=n;i++)scanf("%lld",&a[i]);
for(i=n;i;i--)s[i]=s[i+1]+a[i];
all=sub=1;
for(i=n-1;i;i--){
all=1LL*all*lucas(s[i],a[i])%P;
sub=1LL*sub*lucas(s[i]-1,s[i+1])%P;
}
all=all-sub;
all%=P;
all+=P;
all%=P;
printf("%d",all);
}
I. Boring problem
显然两维独立，每一维分别计算幂和然后相乘即可。
#include<cstdio>
typedef __int128 lll;
long long P=1000000008LL;
lll cal(lll n){
lll A=n*(n+1)*(n+1)/2%P;
lll B=n*(n+1)*(n*2+1)/6%P;
A-=B;
A=A+P;
return A%P;
}
int main(){
int n,m;
scanf("%d%d",&n,&m);
lll A=cal(n),B=cal(m);
A=A*B%P;
A=(A+P)%P;
long long ans=A;
printf("%lld",ans);
}
J. Equation
留坑。
K. Game
势能分析可以发现结果只和最终分法有关而和分解顺序无关，故设$f[i][j]$表示最后一次划分在$i$，总得分为$j$是否可能，bitset优化转移。
时间复杂度$O(\frac{n^4}{64})$。
#include<cstdio>
#include<bitset>
using namespace std;
const int N=305,M=N*N/2;
int n,m,i,j;bitset<M>f[N];
int main(){
scanf("%d%d",&n,&m);
if(m==0)return puts("YES"),0;
if(m>=M)return puts("NO"),0;
for(i=1;i<n;i++){
f[i][i*(n-i)]=1;
for(j=i+1;j<n;j++)f[j]|=f[i]<<((j-i)*(n-j));
}
for(i=1;i<=n;i++)if(f[i][m])return puts("YES"),0;
puts("NO");
}
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