Problem A math
Based on the number of functions $ f (x) $ represents the $ x (x ∉ Prime) $ is the second largest factor,
Given $ l, r $ obtained $ \ sum \ limits_ {i = l, i ∉ Prime} ^ rf (i) $.
For $ 100 \% $ data, $ 1 \ leq l \ leq r \ leq 5 \ times 10 ^ 9 $
Solution: We think for $ rl \ leq 10 ^ 7 $ how to deal with,
Obviously, $ f (x) = \ frac {x} {d_ {min} (x)} $
Therefore, we can use $ \ sqrt {5 \ times 10 ^ 9} $ to a digital number in the screen section, to identify these numbers $ d_ {min} (x) $
In this way, we can solve the problem in the complexity of $ O (rl) $ a.
However, for $ 100 \% $ of data is not enough to pass, so we consider playing table locally, at intervals of $ 10 ^ 7 $ hit list.
So, hit the block table generated AC program was born.
#include <bits/stdc++.h> using namespace std; typedef long long ll; const int N = 100005; ll a[505] = {0,16504974832918ll,49514807041302ll,82524641299076ll,115534479934505ll,148544254405234ll,181554158308885ll,214563872594611ll,247573764207188ll,280583538103016ll,313593361626057ll,346602992946568ll,379613110929775ll,412622718262607ll,445632601916252ll,478642437755379ll,511652194457614ll,544661947496041ll,577671637940336ll,610681752050814ll,643691404588574ll,676701244176651ll,709711054775594ll,742720800286221ll,775730381310093ll,808740585896190ll,841750243027296ll,874760117073666ll,907769696410853ll,940779525226536ll,973789738206002ll,1006799160955474ll,1039808877324861ll,1072819018026025ll,1105828761150350ll,1138838541727837ll,1171848079131282ll,1204858104382139ll,1237867536035320ll,1270877842610423ll,1303887672241912ll,1336897155489979ll,1369907175171416ll,1402916647740933ll,1435927028696641ll,1468936230981939ll,1501946419251818ll,1534955574922012ll,1567966183816364ll,1600975595444924ll,1633985443056901ll,1666995678674159ll,1700005364180681ll,1733014763642307ll,1766024840505475ll,1799034631307151ll,1832044002952252ll,1865054030679522ll,1898064068452051ll,1931073520776903ll,1964083598976891ll,1997093098588907ll,2030103003059564ll,2063112702217827ll,2096122737310551ll,2129133054850411ll,2162142198328595ll,2195152115983905ll,2228161264785474ll,2261172544708734ll,2294181378777450ll,2327191233379424ll,2360201373720367ll,2393211269318476ll,2426220486575075ll,2459230819150224ll,2492240046752302ll,2525251064525340ll,2558259845985451ll,2591268636654841ll,2624280077930602ll,2657288734859772ll,2690299477550963ll,2723308150991146ll,2756319304183776ll,2789328790852636ll,2822338626083272ll,2855347891381860ll,2888357969641819ll,2921367932058355ll,2954377671380517ll,2987387687179118ll,3020395919679682ll, 3053407972625776ll, 3086417613231907ll, 3119426240525339ll, 3152436403285179ll, 3185446530826116ll, 3218455004607940ll, 3251466542904717ll, 3284474102416981ll, 3317485456042933ll, 3350494835559564ll, 3383505323561767ll, 3416514694107384ll, 3449524616389762ll, 3482535136132263ll, 3515544067001265ll, 3548552960601121ll, 3581563404252177ll, 3614573730318868ll, 3647583278380671ll, 3680593224979870ll, 3713603049026829ll, 3746614512962914ll, 3779623637711295ll, 3812631445796912ll, 3845642667207059ll, 3878650989452908ll, 3911661656239387ll, 3944671816767424ll, 3977681760271560ll, 4010691144633474ll, 4043702388095629ll, 4076710028369395ll, 4109720346259831ll, 4142730350170032ll, 4175739772832184ll, 4208750686895433ll, 4241759731456393ll, 4274769641755841ll, 4307777872223390ll, 4340789892971559ll, 4373799166736030ll, 4406809512953890ll, 4439817532702298ll, 4472829307111790ll,4505836643327275ll,4538848618830486ll,4571857797045020ll,4604867900156142ll,4637878508493697ll,4670886796634050ll,4703896752766576ll,4736906975045792ll,4769917026147829ll,4802926090607880ll,4835937748578967ll,4868944057292098ll,4901955388347787ll,4934964803726201ll,4967976054354879ll,5000985142582223ll,5033994769269421ll,5067004519698299ll,5100015475381224ll,5133023523525477ll,5166035548680638ll,5199044102202040ll,5232053769509974ll,5265063976719498ll,5298071256992732ll,5331086032306110ll,5364093009886904ll,5397101286590264ll,5430112363775912ll,5463125358224685ll,5496130321891579ll,5529143222333432ll,5562151266937027ll,5595161916107909ll,5628170955930451ll,5661180935315217ll,5694191463640877ll,5727200867537917ll,5760210082013019ll,5793220944569250ll,5826231266751643ll,5859240047889761ll,5892248661408006ll,5925259731715720ll,5958268524424498ll,5991280302232593ll,6024290558811088ll,6057297413541092ll,6090308767576979ll,6123319740301532ll,6156326709064649ll,6189337531550686ll,6222350666231978ll,6255356723766425ll,6288367605824763ll,6321377226929040ll,6354387848727584ll,6387396476006881ll,6420406295019013ll,6453418361894346ll,6486425279806916ll,6519434933117769ll,6552447537063411ll,6585455288301656ll,6618465571313108ll,6651473219530721ll,6684487597609615ll,6717493272112836ll,6750503897921972ll,6783514370819588ll,6816524779187087ll,6849533841224964ll,6882543200522966ll,6915555567048002ll,6948563176458473ll,6981569435885717ll,7014587121946570ll,7047593188567860ll,7080603174566593ll,7113612266982300ll,7146622391827726ll,7179631332633195ll,7212642629393380ll,7245652465198424ll,7278661503882743ll,7311671230143188ll,7344681060552161ll,7377691956846305ll,7410698365443030ll,7443713861235034ll,7476716796858509ll, 7509729721125427ll, 7542738638892030ll, 7575747728275743ll, 7608761023356292ll, 7641769632981677ll, 7674779747704999ll, 7707788455568150ll, 7740801037220385ll, 7773805775344854ll, 7806820992857333ll, 7839827911666137ll, 7872836137053706ll, 7905848048044634ll, 7938859068637500ll, 7971867868233332ll, 8004875132181171ll, 8037889423443128ll, 8070896110232830ll, 8103906084599451ll, 8136918977662318ll, 8169925767524876ll, 8202934185046077ll, 8235946072038061ll, 8268954361637275ll, 8301967510589725ll, 8334974643904362ll, 8367982557462832ll, 8400994813545019ll, 8434004890087843ll, 8467013576718783ll, 8500020663978651ll, 8533037879707351ll, 8566043768970011ll, 8599053931450993ll, 8632062998388608ll, 8665072097208580ll, 8698085691961056ll, 8731093766071517ll, 8764101245824907ll, 8797112666888496ll, 8830121449268148ll, 8863131724440902ll, 8896143109833470ll, 8929152677519312ll,8962160797626791ll,8995170318618191ll,9028182599892397ll,9061189984462759ll,9094202374215214ll,9127209986538911ll,9160218609463869ll,9193230004788814ll,9226239318764918ll,9259249078019552ll,9292258777511949ll,9325269472475613ll,9358278358670667ll,9391290089218066ll,9424298477932490ll,9457306969271855ll,9490319056893750ll,9523328200440578ll,9556338086313235ll,9589345316007227ll,9622357620373676ll,9655368897041442ll,9688375426730456ll,9721388824503967ll,9754397928198984ll,9787402454672295ll,9820418873814550ll,9853427280754508ll,9886434987872625ll,9919445137206151ll,9952458047898686ll,9985464773711174ll,10018477222500475ll,10051480721951374ll,10084498424737300ll,10117503035880454ll,10150511283380008ll,10183525514995947ll,10216533219700575ll,10249545115462676ll,10282555420096173ll,10315560982306657ll,10348573391185967ll,10381583589631280ll,10414590731461579ll,10447602956176541ll,10480610763020224ll,10513619676792520ll,10546634205927295ll,10579639799250750ll,10612656106927603ll,10645658290948488ll,10678673831958315ll,10711680860306978ll,10744688101131573ll,10777702974875348ll,10810713172243588ll,10843718017801528ll,10876727997999731ll,10909742976356068ll,10942745552837649ll,10975759761937138ll,11008768835479689ll,11041780044438665ll,11074792557248863ll,11107797713588017ll,11140805571789883ll,11173816471269923ll,11206833772230056ll,11239831597194058ll,11272848520003779ll,11305859986584876ll,11338861243119995ll,11371876909802235ll,11404891745082435ll,11437892311766835ll,11470908161801780ll,11503914618689504ll,11536926888562700ll,11569930981965041ll,11602948747128948ll,11635953845463421ll,11668964037083148ll,11701974936299703ll,11734983722393772ll,11767999922030689ll,11801002552735713ll,11834012757000273ll,11867024891777099ll,11900034039336811ll,11933044843810804ll,11966053496685306ll,11999059688845505ll,12032073112983797ll,12065079677645057ll,12098093861265145ll,12131099246321317ll,12164116113639448ll,12197123648653031ll,12230130206829220ll,12263140303351556ll,12296149675284303ll,12329166546109808ll,12362163995178362ll,12395183521502911ll,12428191025607568ll,12461197969424400ll,12494210960510352ll,12527223649818333ll,12560226746125306ll,12593237474958756ll,12626250442185141ll,12659255486596395ll,12692269045326660ll,12725280923267732ll,12758287872178920ll,12791300643783440ll,12824306072686417ll,12857318478711430ll,12890328450655256ll,12923342576879840ll,12956346521609668ll,12989354995382276ll,13022368136035560ll,13055375395557624ll,13088387187085464ll,13121392049733620ll,13154414533694357ll,13187414264095104ll,13220422590606727ll,13253434148574342ll,13286449933283688ll,13319453431008458ll,13352467789104963ll,13385472713638835ll,13418484508815598ll,13451496706837476ll,13484505638592382ll,13517515572237834ll,13550521844051264ll,13583534160503278ll,13616544283380142ll,13649549091999822ll,13682567242501910ll,13715571164122745ll,13748585740439504ll,13781593375245859ll,13814596002035274ll,13847616023505842ll,13880624709504762ll,13913623628459146ll,13946645941528622ll,13979654150914934ll,14012656937653649ll,14045669725564142ll,14078684346649165ll,14111685964032611ll,14144700787704032ll,14177709247364616ll,14210718916247011ll,14243734152923032ll,14276736945810726ll,14309744840548250ll,14342760538177797ll,14375768115529314ll,14408777643625614ll,14441784467373423ll,14474800518213737ll,14507806709670214ll,14540822608872175ll,14573824919746354ll,14606839372926299ll,14639849842420486ll,14672858257836917ll,14705862325475290ll,14738881008363594ll,14771885160171399ll,14804896733975004ll,14837906464878693ll,14870916255587693ll,14903917659481655ll,14936947636033735ll,14969940694692679ll,15002951189604675ll,15035967606869556ll,15068968755050047ll,15101989631801975ll,15134994655079856ll,15168005036122695ll,15201009986586199ll,15234028811592968ll,15267031260187832ll,15300045501993724ll,15333049338537562ll,15366062641352998ll,15399073651356570ll,15432082313085917ll,15465089073859934ll,15498098318094652ll,15531118673527436ll,15564119081959673ll,15597131003731861ll,15630136078225767ll,15663155107037699ll,15696160746682461ll,15729168956086495ll,15762183829610629ll,15795182488686761ll,15828210435180516ll,15861208151451320ll,15894213750900732ll,15927236153583087ll,15960235870043580ll,15993247157265286ll,16026259644857776ll,16059270353562328ll,16092274734756367ll,16125292189907530ll,16158300602354326ll,16191300181887807ll,16224324519050731ll,16257330916482853ll,16290331056290523ll,16323345837649990ll,16356351621749496ll,16389374934257625ll,16422373187347681ll,16455390970353283ll,16488403156976175ll}; ll l, r; ll _div[10000005]; bool nt_prime[N]; int prime_cnt; int prime[N]; void get_prime(int n) { for (int i = 2; i <= n; i++) { if (!nt_prime[i]) prime[++ prime_cnt] = i; for (int j = 1; j <= prime_cnt && prime[j] * i <= n; j++) { nt_prime[i * prime[j]] = 1; if (i % prime[j] == 0) break; } } } ll calc(ll l, ll r) { memset(_div, 0, sizeof _div); ll res = 0; for (ll i = 1; i <= prime_cnt; i++) { for (ll j = (l - 1) / prime[i] * prime[i] + prime[i]; j <= r; j += prime[i]) { if (!_div[j - l]) _div[j - l] = prime[i]; } } for (ll i = l; i <= r; i++) { if (_div[i - l] && _div[i - l] != i) res += i / _div[i - l]; } return res; } ll solve(ll x) { ll res = 0; ll w = x / 10000000; for (int i = 1; i <= w; i++) res += a[i]; res += calc(w * 10000000 + 1, x); return res; } int main() { get_prime(1e5); scanf("%lld%lld", &l, &r); printf("%lld\n", solve(r) - solve(l - 1)); return 0; }
Problem B false questions
Containing a given n-$ $ tree nodes, each edge has a right side. Maximize tree midpoint set points in the set so that the pairwise distance is greater than or equal to the given L $ $
For $ 100 \% $ data satisfies $ 1 \ leq n \ leq 5 \ times 10 ^ 5,1 \ leq L, w \ leq 10 ^ 9 $
Randomly generated data guarantee
Solution: depth from $ 1 $ run tree, greedy, depth greater priority, once removed dfs current new added point set a point near the point $ $ L.
For random data, it should be $ complexity O (can live) $
#pragma GCC optimize(3) #include <bits/stdc++.h> #define int long long using namespace std; const int N=5e5+10; struct rec{ int pre,to,w; }a[N<<1]; int d[N],n,l,head[N],tot,p[N]; bool vis[N]; bool cmp(int a,int b){return d[a]>d[b];} inline int read() { int X=0,w=0; char c=0; while(c<'0'||c>'9') {w|=c=='-';c=getchar();} while(c>='0'&&c<='9') X=(X<<3)+(X<<1)+(c^48),c=getchar(); return w?-X:X; } void adde(int u,int v,int w) { a[++tot].for =head[u]; a[tot].to=v; a[tot].w=w; head[u]=tot; } void dfs1(int u,int fa) { for (int i=head[u];i;i=a[i].pre) { int v=a[i].to; if (v==fa) continue; d[v]=d[u]+a[i].w; dfs1(v,u); } } void dfs2(int u,int fa,int L) { if (L<=0) return; vis[u]=1; for (int i=head[u];i;i=a[i].pre) { int v=a[i].to; if (v==fa) continue; dfs2(v,u,L-a[i].w); } } signed main() { n=read();l=read(); for (int i=1;i<=n-1;i++) { int u=read(),v=read(),w=read();//scanf("%d%d%d",&u,&v,&w); adde(u,v,w); adde(v,u,w); } dfs1(1,0); for (int i=1;i<=n;i++) p[i]=i; sort(p+1,p+1+n,cmp); int ans = 0; for (int i=1;i<=n;i++) if (!vis[p[i]]) dfs2(p[i],0,l),ans++; printf("%lld\n",ans); return 0; }
Problem C Counting Problem
Given a sequence $ a_i (1 \ leq i \ leq n) $, $ determined sequence number b_i (1 \ leq i \ leq 2n) $ meets the following limitations:
1. For any $ 1 \ leq i \ leq n $, has $ b_i | b_ {i + n} $ and $ b_ {i + n} | a_i $
2. $\prod\limits_{i=n+1}^{2n} b_i \leq \prod\limits_{i=1}^{n} {b_i}^2$
The output in the sense of mold $ 998,244,353 $.
For $ 100 \ $ data satisfies%, $ 1 \ leq n \ leq 100,1 \ leq a_i \ leq 10 ^ 9 $
Solution :
Obviously, when the division stage, $ i $ and $ i + n $ is a whole, indivisible.
If a_i $ $ sufficiently small, we can directly credited $ f [i] [j] $ $ I represents the current into a first and a second $ $ i + n $ b-product removal sequence of the second half of the product of the first half of the sequence b the business plan for the number of $ j $.
Thus such a state of violence and DP, is a $ O (n \ prod a_i \ sqrt {a_i}) $ wonderful algorithm to space complexity level factorial.
Obviously, this number exploded state, we can not be that multiplies the value obtained credited to the state to go, then we will think how tired will not remember to take additional states to go.
Soon, we will find no way to do, this state had in mind.
Observed a sharp $ a_i = 2 ^ k $ divided portions, only one factor! Additional state obviously can use to optimize the number to $ log_2 a_i $ level.
Due to the symmetry factor, i.e. $ \ sqrt {a_i} $ as a symmetrical axis, is greater than $ \ sqrt {a_i} $ less than $ \ sqrt {a_i} $ number of factors are the same.
Since the number of selection are independent, non-interfering, it satisfies $ \ prod \ limits_ {i = n + 1} ^ {2n} b_i \ leq \ prod \ limits_ {i = 1} ^ {n} {b_i } ^ 2 $ and satisfy $ \ sequence number prod \ limits_ {i = n + 1} ^ {2n} b_i \ geq \ prod \ limits_ {i = 1} ^ {n} {b_i} ^ 2 $ are the same.
For the scheme does not consider a second limit number, relatively easy to find, is $ \ prod \ frac {(c + 1) (c + 2)} {2} $, where $ c $ represents a number, a certain factor the number of occurrences.
Thus, we need to consider exactly, $ \ prod \ limits_ {i = n + 1} ^ {2n} b_i = \ prod \ limits_ {i = 1} ^ {n} number {b_i} ^ 2 $ solutions.
For each individual factor, as we can deal with in accordance with the $ 2 $ (logarithmic denoted state), if the current number is not a multiple thereof, then skip, continue following the transfer.
Finally, the answer to each separate quality factor is multiplied by the final answer.
$ $ ANS1 provided without regard to the number of sequences showing a second embodiment of limitation, $ ans2 $ represents a number satisfying $ \ prod \ limits_ {i = n + 1} ^ {2n} b_i = \ prod \ limits_ {i = 1} ^ number sequence {n} {b_i} ^ 2 $ a.
The final answer is $ \ frac {ans1 + ans2} {2} $
Specific dynamic programming equation is quite classic nor what needs special attention.
Complexity should be $ O (Prime \ _Num \ times n \ times \ sum_ {i = 1} ^ {n} log a_i \ times {log_2} ^ 2 Max \ {a_i \})) $
# include <bits/stdc++.h> # define int long long using namespace std; const int N=105,mo=998244353; int f[N][6001],n,a[N],rec[N],t[N]; bool is_pr[31623]; int pr[31623]; vector<int>v; int Pow(int x,int n) { int ans = 1; while (n) { if (n&1) ans=ans*x%mo; x=x*x%mo; n>>=1; } return ans%mo; } void EouLaShai(int Lim) { memset(is_pr,true,sizeof(is_pr)); is_pr[1]=false; for (int i=2;i<=Lim;i++) { if (is_pr[i]) pr[++pr[0]]=i; for (int j=1;j<=pr[0]&&i*pr[j]<=Lim;j++) { is_pr[i*pr[j]]=false; if (i%pr[j]==0) break; } } } signed main() { int inv2=Pow(2,mo-2); EouLaShai(31622); int ret1=1; scanf("%lld",&n); for (int i=1;i<=n;i++) scanf("%lld",&a[i]); memcpy(rec,a,sizeof(a)); for (int i=1;i<=n;i++) { for (int j=1;j<=pr[0];j++) { if (pr[j]>rec[i]) break; if (rec[i]%pr[j]==0) { v.push_back(pr[j]); int c=0; while (rec[i]>1 && rec[i]%pr[j]==0) rec[i]/=pr[j],c++; ret1=ret1*((c+1)*(c+2)/2)%mo; } } if (rec[i]!=1) ret1=ret1*((1+1)*(1+2)/2)%mo,v.push_back(rec[i]); } sort(v.begin(),v.end()); v.erase(unique(v.begin(),v.end()),v.end()); int ret2=1; for (int tmp=0;tmp<v.size();tmp++) { int num = v[tmp],sum=0; for (int i=1;i<=n;i++) { t[i]=0; int mht=a[i]; while (mht>1&&mht%num==0) mht/=num,t[i]++; sum+=t[i]; } for (int i=0;i<=n;i++) for (int j=-sum;j<=sum;j++) f[i][j+3000]=0; f[0][0+3000]=1; for (int i=0;i<=n-1;i++) for (int j=-sum;j<=sum;j++) if (f[i][j+3000]) { if (a[i+1]%num!=0) { f[i+1][j+3000]=f[i][j+3000]; continue;} int lim = t[i+1]; for (int k=0;k<=lim;k++) for (int w=0;w<=k;w++) if (j+k-2*w>=-sum &&j+k-2*w<=sum) { f[i+1][j+k-2*w+3000]=(f[i+1][j+k-2*w+3000]+f[i][j+3000])%mo; } } ret2=ret2*f[n][0+3000]%mo; } int ans = (ret1+ret2)*inv2%mo; printf("%lld\n",ans); return 0; }