蓝桥杯 1223 第 2 场 小白入门赛

蓝桥小课堂-平方和

• 模拟 1 2 + 2 2 + 3 2 + ⋯ + n 2 = n    ⋅    ( n + 1 )    ⋅    ( 2 n + 1 ) 6 1^2+2^2+3^2+\cdots+n^2=\dfrac{n\;\cdot\;(n +1)\;\cdot\;(2n+1)}{6} 12+22+32+⋯+n2=6n⋅(n+1)⋅(2n+1)。

 write(n * (n + 1) * (n * 2 + 1) / 6);

房顶漏水啦

• m a x ( 最大的行 − 最小的行 , 最大的列 − 最小的列 ) + 1 max(最大的行-最小的行,最大的列-最小的列) + 1 max(最大的行−最小的行,最大的列−最小的列)+1

ps: 2 ≤ n ≤ 1 0 10 2\le n \le10^{10} 2≤n≤1010

signed main() {
    int T = 1;
//    T = read();
    while (T--) {
        int n = read(), m = read();
        int col1 = 1e12, col2 = -1e12, row1 = 1e12, row2 = -1e12;
        while (m--) {
            int u = read(), v = read();
            col1 = min(col1, u), col2 = max(col2, u);
            row1 = min(row1, v), row2 = max(row2, v);
        }
        write(max(col2 - col1 + 1, row2 - row1 + 1));
    }
    return 0;
}

质数王国

• 思路：对每一个数找到其左右两个素数，最小操作次数就是较小差值。
• 欧拉筛筛出 [ 1    ,    1 0 5 + 10 ] [1\;,\;10^5+10] [1,105+10] （范围大于 1 0 5 10^5 105 你不能保证右边的素数不超过 1 0 5 10^5 105）的质数装入 s e t set set。
• 二分找较小差值。

signed main() {
    int T = 1;
//    T = read();
    while (T--) {
        int n = read();
        vector<int> a(n + 1);
        for (int i = 1; i <= n; ++i) a[i] = read();
        vector<bool> isprime(N, false);
        vector<int> prime(N);
        set<int> s;
        int cnt = 0;
        for (int i = 2; i <= N; ++i) {
            if (!isprime[i]) prime[++cnt] = i, s.insert(i);
            for (int j = 1; j <= cnt && i * prime[j] <= N; ++j) {
                isprime[i * prime[j]] = 1;
                if (!(i % prime[j])) break;
            }
        }
        int ans = 0;
        for (int i = 1; i <= n; ++i) {
            if (s.count(a[i])) continue;
            auto it = s.lower_bound(a[i]);
            int l, r;
            if (it == s.end()) r = INF;
            else r = *it;
            if (it == s.begin()) l = -INF;
            else l = *prev(it);
            ans += min(r - a[i], a[i] - l);
        }
        write(ans);
    }
    return 0;
}

取余

• 思路：枚举 b i b_i bi 。以 b i b_i bi 为模一定能将 A A A 分成 A    /    b i + 1 A\;/\;b_i+1 A/bi+1 份。
  
• 如上图可以清晰的看出答案。只需要取下面与 [ S    ,    T ] [S\;,\;T] [S,T] 重合的区间长度即可。你可以将其分成三部分思考（首，中间，尾）。
• 对于 b i = 1 b_i =1 bi=1 的情况，只有当 S = 0 S = 0 S=0 时才有解，此时应该加上 A A A 。

signed main() {
    int T = 1;
//    T = read();
    while (T--) {
        int a = read(), b = read(), s = read(), t = read();
        int ans = 0;
        for (int i = 2; i <= b; ++i) {
            if (s > i - 1 || a < s) continue;
            int p = a / i + 1;
            if (p > 2) {
                ans += (p - 1) * (min(t, i - 1) - s + 1) - (s == 0);
                int cnt = a - (i - 1) - (p - 2) * i;
                if (cnt - 1 >= s) ans += min(t, cnt - 1) - s + 1;
            }
            else if (p == 2) {
                ans += min(t, i - 1) - s + 1 - (s == 0);
                int cnt = a - (i - 1);
                if (cnt - 1 >= s) ans += min(t, cnt - 1) - s + 1;
            }
            else ans += min(t, a) - s + 1 - (s == 0);
        ans += (s == 0) * a;
        write(ans);
    }
    return 0;
}

数学尖子生

• 思路：容斥。
• 不能被 a a a 整除，但一定能被 1 , 2 , ⋯   , 3 , a − 1 1,2,\cdots,3,a-1 1,2,⋯,3,a−1 整除。所以这个数一定是 1 , 2 , ⋯   , 3 , a − 1 1,2,\cdots,3,a-1 1,2,⋯,3,a−1 最小公倍数的倍数，考虑容斥，用 1 , 2 , ⋯   , 3 , a − 1 1,2,\cdots,3,a-1 1,2,⋯,3,a−1 最小公倍数的倍数的个数 − - − 1 , 2 , ⋯   , 3 , a − 1 , a 1,2,\cdots,3,a-1,a 1,2,⋯,3,a−1,a 最小公倍数的倍数的个数。

int gcd(int a, int b) {return b? gcd(b, a % b): a;}
int lcm(int a, int b) {return a * b / gcd(a, b);}
signed main() {
    int T = 1;
    T = read();
    vector<int> p(N);
    int cnt = 0;
    p[cnt] = 1;
    while (p[cnt] < 1e16) {
        ++cnt;
        p[cnt] = lcm(p[cnt - 1], cnt);
    }
    while (T--) {
        int a = read(), n = read();
        if (a > cnt) puts("0");
        else writeln(n / p[a - 1] - n / p[a]);
    }
    return 0;
}

魔术师

• 思路：线段树 + 矩阵
• 举例： [    a    ,    b    ,    c    ] [\;a\;,\;b\;,\;c\;] [a,b,c]
• • o p t 1    a    b opt_1\;a\;b opt1ab ： [    a b c    ] × [    0 1 0       1 0 0       0 0 1    ] \begin{bmatrix}\;a\quad b\quad c\;\end{bmatrix}\times\begin{bmatrix}\;0\quad 1\quad 0\;\\\;1\quad 0\quad 0\;\\\;0\quad 0\quad 1\;\end{bmatrix} [abc]× 010100001
• • o p t 2    a    b opt_2\;a\;b opt2ab ： [    a b c    ] × [    0 0 0       1 1 0       0 0 1    ] \begin{bmatrix}\;a\quad b\quad c\;\end{bmatrix}\times\begin{bmatrix}\;0\quad 0\quad 0\;\\\;1\quad 1\quad 0\;\\\;0\quad 0\quad 1\;\end{bmatrix} [abc]× 000110001
• • o p t 3    a    b opt_3\;a\;b opt3ab ： [    a b c    ] × [    2 0 0       0 1 0       0 0 1    ] \begin{bmatrix}\;a\quad b\quad c\;\end{bmatrix}\times\begin{bmatrix}\;2\quad 0\quad 0\;\\\;0\quad 1\quad 0\;\\\;0\quad 0\quad 1\;\end{bmatrix} [abc]× 200010001

ps：要用懒标记，其实和一般线段树下放懒标记没太大区别，第一次接触可能需要自己体会一下。

#include <bits/stdc++.h>
#define il inline
#define get getchar
#define put putchar
#define is isdigit
#define int long long
#define dfor(i,a,b) for(int i=a;i<=b;++i)
#define dforr(i,a,b) for(int i=a;i>=b;--i)
#define dforn(i,a,b) for(int i=a;i<=b;++i,put(10))
#define mem(a,b) memset(a,b,sizeof(a))
#define memc(a,b) memcpy(a,b,sizeof(a))
#define pr 114514191981
#define gg(a) cout<<a,put(32)
#define INF 0x7fffffff
#define tt(x) cout<<x<<'\n'
#define tf(x) cout << "-> " << x << " <-" << '\n';
#define endl '\n'
#define ls i<<1
#define rs i<<1|1
#define la(r) tr[r].ch[0]
#define ra(r) tr[r].ch[1]
#define lowbit(x) (x&-x)
#define ct cin.tie(nullptr),ios_base::sync_with_stdio(false)
using namespace std;
typedef unsigned int ull;
typedef pair<int, int> pii;
int read(void) {
    int x=0,f=1;char c=get();
    while(!is(c)) (f=c==45?-1:1),c=get();
    while(is(c)) x=(x<<1)+(x<<3)+(c^48),c=get();
    return x*f;
}
void write(int x) {
    if (x < 0) x = -x, put(45);
    if (x > 9) write(x / 10);
    put((x % 10) ^ 48);
}
#define writeln(a) write(a), put(10)
#define writesp(a) write(a), put(32)
#define writessp(a) put(32), write(a)
const int N = 1e5 + 10, M = 2e5 + 10, SN = 1e3 + 10, mod = 1e8, MOD = 998244353;
int c[N], a[N << 2][3], base[3][3], lazy[N << 2][3][3];
void init(int p[3][3]) {
    for (int i = 0; i < 3; ++i) {
        for (int j = 0; j < 3; ++j) p[i][j] = i == j;
    }
}
void pushup(int i, int a[][3]) {
    for (int j = 0; j < 3; ++j) a[i][j] = (a[ls][j] + a[rs][j]) % MOD;
}
void pushdown1(int a[3][3], int b[3][3]) {
    int t[3][3];
    mem(t, 0);
    for (int i = 0; i < 3; ++i) {
        for (int j = 0; j < 3; ++j) {
            for (int k = 0; k < 3; ++k) t[i][j] = (t[i][j] + a[i][k] * b[k][j]) % MOD;
        }
    }
    for (int i = 0; i < 3; ++i) {
        for (int j = 0; j < 3; ++j) a[i][j] = t[i][j];
    }
}
void pushdown2(int a[3], int b[3][3]) {
    int t[3];
    mem(t, 0);
    for (int i = 0; i < 3; ++i) {
        for (int j = 0; j < 3; ++j) t[i] = (t[i] + a[j] * b[j][i]) % MOD;
    }
    for (int i = 0; i < 3; ++i) a[i] = t[i];
}
void build(int i, int l, int r) {
    init(lazy[i]);
    if (l == r) { a[i][c[l] - 1] = 1; return ;}
    int mid = (l + r) >> 1;
    build(ls, l, mid), build(rs, mid + 1, r);
    pushup(i, a);
}
void modify(int i, int l, int r, int L, int R) {
    if (L <= l && r <= R) {
        pushdown1(lazy[i], base);
        pushdown2(a[i], base);
        return ;
    }
    pushdown1(lazy[i << 1], lazy[i]), pushdown1(lazy[i << 1 | 1], lazy[i]);
    pushdown2(a[ls], lazy[i]), pushdown2(a[rs], lazy[i]);
    init(lazy[i]);
    int mid = (l + r) >> 1;
    if (L <= mid) modify(ls, l, mid, L, R);
    if (R > mid) modify(rs, mid + 1, r, L, R);
    pushup(i, a);
}
signed main() {
    int T = 1;
//    T = read();
    while (T--) {
        int n = read(), m = read();
        for (int i = 1; i <= n; ++i) c[i] = read();
        build(1, 1, n);
        while (m--) {
            int l = read(), r = read(), op = read(), cla = read() - 1;
            init(base);
            if (op == 3) {
                base[cla][cla] = 2;
            }
            else {
                int clb = read() - 1;
                if (op & 1) {
                    base[cla][cla] = base[clb][clb] = 0;
                    base[cla][clb] = base[clb][cla] = 1;
                }
                else base[cla][cla] = 0, base[cla][clb] = 1;
            }
            modify(1, 1, n, l, r);
            writesp(a[1][0]), writesp(a[1][1]), writeln(a[1][2]);
        }
    }
    return 0;
}