#line 1 "/home/maspy/compro/library/my_template.hpp" #if defined(LOCAL) #include #else #pragma GCC optimize("Ofast") #pragma GCC optimize("unroll-loops") #include using namespace std; using ll = long long; using u32 = unsigned int; using u64 = unsigned long long; using i128 = __int128; using u128 = unsigned __int128; using f128 = __float128; template constexpr T infty = 0; template <> constexpr int infty = 1'000'000'000; template <> constexpr ll infty = ll(infty) * infty * 2; template <> constexpr u32 infty = infty; template <> constexpr u64 infty = infty; template <> constexpr i128 infty = i128(infty) * infty; template <> constexpr double infty = infty; template <> constexpr long double infty = infty; using pi = pair; using vi = vector; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using vvvvvc = vector>; template using pq = priority_queue; template using pqg = priority_queue, greater>; #define vv(type, name, h, ...) \ vector> name(h, vector(__VA_ARGS__)) #define vvv(type, name, h, w, ...) \ vector>> name( \ h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name( \ a, vector>>( \ b, vector>(c, vector(__VA_ARGS__)))) // https://trap.jp/post/1224/ #define FOR1(a) for (ll _ = 0; _ < ll(a); ++_) #define FOR2(i, a) for (ll i = 0; i < ll(a); ++i) #define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i) #define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c)) #define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i) #define overload4(a, b, c, d, e, ...) e #define overload3(a, b, c, d, ...) d #define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__) #define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__) #define FOR_subset(t, s) \ for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s))) #define all(x) x.begin(), x.end() #define len(x) ll(x.size()) #define elif else if #define eb emplace_back #define mp make_pair #define mt make_tuple #define fi first #define se second #define stoi stoll int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(ll x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } int popcnt_mod_2(int x) { return __builtin_parity(x); } int popcnt_mod_2(u32 x) { return __builtin_parity(x); } int popcnt_mod_2(ll x) { return __builtin_parityll(x); } int popcnt_mod_2(u64 x) { return __builtin_parityll(x); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template T floor(T a, T b) { return a / b - (a % b && (a ^ b) < 0); } template T ceil(T x, T y) { return floor(x + y - 1, y); } template T bmod(T x, T y) { return x - y * floor(x, y); } template pair divmod(T x, T y) { T q = floor(x, y); return {q, x - q * y}; } template T SUM(const vector &A) { T sm = 0; for (auto &&a: A) sm += a; return sm; } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define LB(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define UB(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define UNIQUE(x) \ sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit() template T POP(deque &que) { T a = que.front(); que.pop_front(); return a; } template T POP(pq &que) { T a = que.top(); que.pop(); return a; } template T POP(pqg &que) { T a = que.top(); que.pop(); return a; } template T POP(vc &que) { T a = que.back(); que.pop_back(); return a; } template ll binary_search(F check, ll ok, ll ng, bool check_ok = true) { if (check_ok) assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; (check(x) ? ok : ng) = x; } return ok; } template double binary_search_real(F check, double ok, double ng, int iter = 100) { FOR(iter) { double x = (ok + ng) / 2; (check(x) ? ok : ng) = x; } return (ok + ng) / 2; } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } // ? は -1 vc s_to_vi(const string &S, char first_char) { vc A(S.size()); FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); } return A; } template vector cumsum(vector &A, int off = 1) { int N = A.size(); vector B(N + 1); FOR(i, N) { B[i + 1] = B[i] + A[i]; } if (off == 0) B.erase(B.begin()); return B; } // stable sort template vector argsort(const vector &A) { vector ids(len(A)); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); }); return ids; } // A[I[0]], A[I[1]], ... template vc rearrange(const vc &A, const vc &I) { vc B(len(I)); FOR(i, len(I)) B[i] = A[I[i]]; return B; } #endif #line 1 "/home/maspy/compro/library/other/io.hpp" #define FASTIO #include // https://judge.yosupo.jp/submission/21623 namespace fastio { static constexpr uint32_t SZ = 1 << 17; char ibuf[SZ]; char obuf[SZ]; char out[100]; // pointer of ibuf, obuf uint32_t pil = 0, pir = 0, por = 0; struct Pre { char num[10000][4]; constexpr Pre() : num() { for (int i = 0; i < 10000; i++) { int n = i; for (int j = 3; j >= 0; j--) { num[i][j] = n % 10 | '0'; n /= 10; } } } } constexpr pre; inline void load() { memcpy(ibuf, ibuf + pil, pir - pil); pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin); pil = 0; if (pir < SZ) ibuf[pir++] = '\n'; } inline void flush() { fwrite(obuf, 1, por, stdout); por = 0; } void rd(char &c) { do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); } void rd(string &x) { x.clear(); char c; do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); do { x += c; if (pil == pir) load(); c = ibuf[pil++]; } while (!isspace(c)); } template void rd_real(T &x) { string s; rd(s); x = stod(s); } template void rd_integer(T &x) { if (pil + 100 > pir) load(); char c; do c = ibuf[pil++]; while (c < '-'); bool minus = 0; if constexpr (is_signed::value || is_same_v) { if (c == '-') { minus = 1, c = ibuf[pil++]; } } x = 0; while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; } if constexpr (is_signed::value || is_same_v) { if (minus) x = -x; } } void rd(int &x) { rd_integer(x); } void rd(ll &x) { rd_integer(x); } void rd(i128 &x) { rd_integer(x); } void rd(u32 &x) { rd_integer(x); } void rd(u64 &x) { rd_integer(x); } void rd(u128 &x) { rd_integer(x); } void rd(double &x) { rd_real(x); } void rd(long double &x) { rd_real(x); } void rd(f128 &x) { rd_real(x); } template void rd(pair &p) { return rd(p.first), rd(p.second); } template void rd_tuple(T &t) { if constexpr (N < std::tuple_size::value) { auto &x = std::get(t); rd(x); rd_tuple(t); } } template void rd(tuple &tpl) { rd_tuple(tpl); } template void rd(array &x) { for (auto &d: x) rd(d); } template void rd(vc &x) { for (auto &d: x) rd(d); } void read() {} template void read(H &h, T &... t) { rd(h), read(t...); } void wt(const char c) { if (por == SZ) flush(); obuf[por++] = c; } void wt(const string s) { for (char c: s) wt(c); } void wt(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) wt(s[i]); } template void wt_integer(T x) { if (por > SZ - 100) flush(); if (x < 0) { obuf[por++] = '-', x = -x; } int outi; for (outi = 96; x >= 10000; outi -= 4) { memcpy(out + outi, pre.num[x % 10000], 4); x /= 10000; } if (x >= 1000) { memcpy(obuf + por, pre.num[x], 4); por += 4; } else if (x >= 100) { memcpy(obuf + por, pre.num[x] + 1, 3); por += 3; } else if (x >= 10) { int q = (x * 103) >> 10; obuf[por] = q | '0'; obuf[por + 1] = (x - q * 10) | '0'; por += 2; } else obuf[por++] = x | '0'; memcpy(obuf + por, out + outi + 4, 96 - outi); por += 96 - outi; } template void wt_real(T x) { ostringstream oss; oss << fixed << setprecision(15) << double(x); string s = oss.str(); wt(s); } void wt(int x) { wt_integer(x); } void wt(ll x) { wt_integer(x); } void wt(i128 x) { wt_integer(x); } void wt(u32 x) { wt_integer(x); } void wt(u64 x) { wt_integer(x); } void wt(u128 x) { wt_integer(x); } void wt(double x) { wt_real(x); } void wt(long double x) { wt_real(x); } void wt(f128 x) { wt_real(x); } template void wt(const pair val) { wt(val.first); wt(' '); wt(val.second); } template void wt_tuple(const T t) { if constexpr (N < std::tuple_size::value) { if constexpr (N > 0) { wt(' '); } const auto x = std::get(t); wt(x); wt_tuple(t); } } template void wt(tuple tpl) { wt_tuple(tpl); } template void wt(const array val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } template void wt(const vector val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } void print() { wt('\n'); } template void print(Head &&head, Tail &&... tail) { wt(head); if (sizeof...(Tail)) wt(' '); print(forward(tail)...); } // gcc expansion. called automaticall after main. void __attribute__((destructor)) _d() { flush(); } } // namespace fastio using fastio::read; using fastio::print; using fastio::flush; #define INT(...) \ int __VA_ARGS__; \ read(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ read(__VA_ARGS__) #define U32(...) \ u32 __VA_ARGS__; \ read(__VA_ARGS__) #define U64(...) \ u64 __VA_ARGS__; \ read(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ read(__VA_ARGS__) #define CHAR(...) \ char __VA_ARGS__; \ read(__VA_ARGS__) #define DBL(...) \ double __VA_ARGS__; \ read(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ read(name) #define VV(type, name, h, w) \ vector> name(h, vector(w)); \ read(name) void YES(bool t = 1) { print(t ? "YES" : "NO"); } void NO(bool t = 1) { YES(!t); } void Yes(bool t = 1) { print(t ? "Yes" : "No"); } void No(bool t = 1) { Yes(!t); } void yes(bool t = 1) { print(t ? "yes" : "no"); } void no(bool t = 1) { yes(!t); } #line 3 "main.cpp" void solve() { // P 上限 // X 中央 // Y 最頻値の平均 LL(N, P, X, Y); bool CHECK = (N == 3 && P == 100 && X == 0 && Y == 0); ++P; ll ANS = -1; auto range_sum = [&](ll a, ll b) -> ll { return (a + b - 1) * (b - a) / 2; }; auto to_range = [&](ll S, ll n) -> tuple { if (n == 0) { assert(S == 0); return {0, 0, -1}; } // L+...+(L+(n-1))<=S ll rhs = S - (n - 1) * n / 2; ll L = floor(rhs, n); ll R = L + n; ll ex = range_sum(L, R + 1) - S; assert(L <= ex && ex <= R); if (ex == R) return {L, R, -1}; return {L, R + 1, ex}; }; auto greedy = [&](ll& sm, ll& n, ll L, ll R, ll can) -> void { assert(n >= 0); if (can == 0) return; if ((R - L) * can <= n) { n -= (R - L) * can; sm += range_sum(L, R) * can; return; } ll k = n / can; n -= can * k; sm += range_sum(R - k, R) * can; sm += (R - k - 1) * n; n = 0; }; auto solve_0 = [&](ll n, ll a, ll b) -> void { // X が最頻値でないとき //・最頻値の頻度 n //・上側最頻値の個数 a //・下側最頻値の個数 b if (n == 1) return; if (a + b == 0) return; if (a * n > N / 2) return; if (b * n > N / 2) return; if (a > X || b > P - (X + 1)) return; ll use_X_upper = (N / 2) - (b * n + (P - X - 1 - b) * (n - 1)); chmax(use_X_upper, 0); if (use_X_upper + 1 >= n) return; ll use_X_lower = (N / 2) - (a * n); chmin(use_X_lower, (n - 1) - 1 - use_X_upper); if (N / 2 < use_X_lower + n * a) return; if (N / 2 < use_X_upper + n * b) return; auto F = [&](ll S1) -> void { ll S2 = Y * (a + b) - S1; if (S1 < range_sum(0, a)) return; if (range_sum(X - a, X) < S1) return; if (S2 < range_sum(X + 1, X + 1 + b)) return; if (range_sum(P - b, P) < S2) return; if (a == 0 && S1 != 0) return; if (b == 0 && S2 != 0) return; auto [L1, R1, ex1] = to_range(S1, a); auto [L2, R2, ex2] = to_range(S2, b); if (L1 == R1) L1 = X, R1 = X; if (L2 == R2) L2 = P, R2 = P; ll sm = n * (S1 + S2) + X * (1 + use_X_lower + use_X_upper); ll n1 = N / 2 - use_X_lower - n * a; ll n2 = N / 2 - use_X_upper - n * b; greedy(sm, n1, R1, X, n - 1); if (ex1 != -1) greedy(sm, n1, ex1, ex1 + 1, n - 1); greedy(sm, n1, 0, L1, n - 1); greedy(sm, n2, R2, P, n - 1); if (ex2 != -1) greedy(sm, n2, ex2, ex2 + 1, n - 1); greedy(sm, n2, X + 1, L2, n - 1); // if (CHECK) { print(__LINE__, n, a, b, S1, S2, n1, n2, sm); } if (n1 == 0 && n2 == 0) chmax(ANS, sm); }; vi X1, X2; X1.eb(range_sum(0, a)); X1.eb(range_sum(X - a, X)); if (a != 0) { ll r = N / 2 - use_X_lower - n * a; ll k = r / (n - 1); for (auto& kk: {k - 1, k, k + 1}) X1.eb(range_sum(X - kk - a, X - kk)); } X2.eb(range_sum(X + 1, X + b + 1)); X2.eb(range_sum(P - b, P)); if (b != 0) { ll r = N / 2 - use_X_upper - n * b; ll k = r / (n - 1); for (auto& kk: {k - 1, k, k + 1}) X2.eb(range_sum(X - kk - a, X - kk)); } for (auto& x: X1) F(x); for (auto& x: X2) F(Y * (a + b) - x); }; auto solve_1 = [&](ll n, ll a, ll b) -> void { // X が最頻値である //・最頻値の頻度 n //・上側最頻値の個数 a //・下側最頻値の個数 b // 大体同じことができる if (a * n > N / 2) return; if (b * n > N / 2) return; if (a > X || b > P - (X + 1)) return; if (n == 1) { ll lo = range_sum(0, a) + X + range_sum(X + 1, X + 1 + b); ll hi = range_sum(X - a, X) + X + range_sum(P - b, P); if (lo <= Y * N && Y * N <= hi) { chmax(ANS, Y * N); } return; } // 上側ではあまり X を使いたくない // 下側でなるべく X を消費 ll can_lower = (N / 2) - (a * n); chmin(can_lower, n - 1); ll need_upper = n - 1 - can_lower; chmax(need_upper, (N / 2) - (b * n) - (P - X - b) * (n - 1)); if (need_upper >= n) return; ll use_X_upper = need_upper; ll use_X_lower = n - 1 - use_X_upper; if (N / 2 < use_X_lower + n * a) return; if (N / 2 < use_X_upper + n * b) return; auto F = [&](ll S1) -> void { ll S2 = Y * (a + b + 1) - X - S1; if (S1 < range_sum(0, a)) return; if (range_sum(X - a, X) < S1) return; if (S2 < range_sum(X + 1, X + 1 + b)) return; if (range_sum(P - b, P) < S2) return; if (a == 0 && S1 != 0) return; if (b == 0 && S2 != 0) return; auto [L1, R1, ex1] = to_range(S1, a); auto [L2, R2, ex2] = to_range(S2, b); if (L1 == R1) L1 = X, R1 = X; if (L2 == R2) L2 = P, R2 = P; ll sm = n * (S1 + S2) + X * (1 + use_X_lower + use_X_upper); ll n1 = N / 2 - use_X_lower - n * a; ll n2 = N / 2 - use_X_upper - n * b; greedy(sm, n1, R1, X, n - 1); if (ex1 != -1) greedy(sm, n1, ex1, ex1 + 1, n - 1); greedy(sm, n1, 0, L1, n - 1); greedy(sm, n2, R2, P, n - 1); if (ex2 != -1) greedy(sm, n2, ex2, ex2 + 1, n - 1); greedy(sm, n2, X + 1, L2, n - 1); // if (CHECK) { print(__LINE__, n, a, b, S1, S2, n1, n2, sm); } if (n1 == 0 && n2 == 0) chmax(ANS, sm); }; vi X1, X2; X1.eb(range_sum(0, a)); X1.eb(range_sum(X - a, X)); if (a != 0) { ll r = N / 2 - use_X_lower - n * a; ll k = r / (n - 1); for (auto& kk: {k - 1, k, k + 1}) X1.eb(range_sum(X - kk - a, X - kk)); } X2.eb(range_sum(X + 1, X + b + 1)); X2.eb(range_sum(P - b, P)); if (b != 0) { ll r = N / 2 - use_X_upper - n * b; ll k = r / (n - 1); for (auto& kk: {k - 1, k, k + 1}) X2.eb(range_sum(X - kk - a, X - kk)); } for (auto& x: X1) F(x); for (auto& x: X2) F(Y * (a + b + 1) - X - x); }; FOR(n, 1, N + 1) { FOR(a, N / n + 1) { FOR(b, N / n - a + 1) { solve_0(n, a, b); solve_1(n, a, b); } } } print(ANS); } signed main() { int T = 10; FOR(T) solve(); return 0; }