#include #define rep(i, n) for(ll i = 0; i < ll(n); i++) #define rep2(i, s, n) for(ll i = ll(s); i < ll(n); i++) #define rrep(i, n) for(ll i = ll(n) - 1; i >= 0; i--) #define rrep2(i, n, t) for(ll i = ll(n) - 1; i >= ll(t); i--) #define all(a) a.begin(), a.end() #define SZ(a) ll(a.size()) #define eb emplace_back using namespace std; using ll = long long; using P = pair; using vl = vector; using vvl = vector; using vp = vector

; using vvp = vector; bool chmin(auto& a, auto b) { return a > b ? a = b, 1 : 0; } bool chmax(auto& a, auto b) { return a < b ? a = b, 1 : 0; } using ld = long double; using Point = complex; using std::norm; using vd = vector; using vdd = vector; using pdd = pair; constexpr ld INF = 1e18; constexpr ld EPS = 1e-6; constexpr Point NULL_POINT = (INF, INF); constexpr Point ORIGIN = (0, 0); //reversed priority_queue template class prique :public std::priority_queue, std::greater> {}; inline bool isNullPoint(Point &p){ return p.real() > INF/2; } inline bool equal(const ld &a, const ld &b){ return fabsl(a-b) < EPS; } inline bool equal(const Point &a, const Point &b){ return equal(a.real(), b.real()) && equal(a.imag(), b.imag()); } Point unitVector(const Point &a){ return a / abs(a); } Point normalVector(const Point &a){ return a * Point(0,1); } ld dot(const Point &a, const Point &b){ return a.real() * b.real() + a.imag() * b.imag(); } ld cross(const Point &a, const Point &b){ return (a.real() * b.imag() - a.imag()*b.real()); } Point rotate(const Point &p, const ld &theta){ return p * Point(cos(theta), sin(theta)); } ld radianToDegree(const ld &radian) { return radian * 180.0 / M_PI; } ld degreeToRadian(const ld °ree) { return degree * M_PI / 180.0; } struct Line { Point a, b; Line() = default; Line(Point a, Point b) : a(a), b(b) {} // Ax + By = C Line(ld A, ld B, ld C){ if(equal(A, 0)){ assert(!equal(B, 0)); a = Point(0, C/B), b = Point(1, C/B); } else if(equal(B, 0)){ a = Point(C/A, 0), b = Point(C/A, 1); } else{ a = Point(C/A, 0), b = Point(0, C/B); } } }; struct Segment : Line{ Segment() = default; Segment(Point a, Point b) : Line(a, b){} }; struct Circle { Point p; ld r; Circle() = default; Circle(Point p, ld r) : p(p), r(r) {} bool contain(const Point &q){ return (abs(q-p) <= r+EPS); } }; Point projection(const Line &l, const Point &p){ ld t = dot(l.b-l.a, l.b-p) / norm(l.b-l.a); return l.a + (l.b-l.a) * t; } Point reflection(const Line &l, const Point &p){ return p + (projection(l, p) - p) * ld(2.0); } // a → b → c の位置関係 /** * a → b → c の位置関係 * * @return * 1 : 反時計回り * -1 : 時計回り * 2 : c → a → b * -2 : a → b → c * 0 : a → c → b */ int ccw(const Point &a, Point b, Point c){ b -= a, c -= a; if(cross(b,c) > EPS){ return 1; } else if(cross(b,c) < -EPS){ return -1; } else if(dot(b,c) < 0){ return 2; } else if(norm(b) < norm(c)){ return -2; } return 0; } // 直行? bool isOrthogonal(const Line &a, const Line &b){ return equal(dot(a.b-a.a, b.b-b.a), 0); } // 並行? bool isParallel(const Line &a, const Line &b){ return equal(cross(a.b-a.a, b.b-b.a), 0); } // 交差判定 → 線分 ab に対して c, d が反対側にあるか? bool isIntersect(const Segment &s, const Segment &t){ return ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 && ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0; } //交点 Point crossPoint(const Line &s, const Line &t){ ld d1 = cross(s.b-s.a, t.b-t.a); ld d2 = cross(s.b-s.a, s.b-t.a); if(equal(fabs(d1), 0)){ if(equal(fabs(d2), 0)) return t.a; else return NULL_POINT; } return t.a + (t.b-t.a) * d2 / d1; } Point crossPoint(const Segment &s, const Segment &t){ if(!isIntersect(s, t)) return NULL_POINT; return crossPoint(Line(s), Line(t)); } ld distanceBetweenLineAndPoint(const Line &l, const Point &p){ return fabs(cross(l.b-l.a, p-l.a)) / abs(l.b-l.a); } ld distanceBetweenSegmentAndPoint(const Segment &l, const Point &p){ if(dot(l.b-l.a, p-l.a) < EPS) { return abs(p-l.a); } if(dot(l.a-l.b, p-l.b) < EPS){ return abs(p-l.b); } return fabs(cross(l.b-l.a, p-l.a)) / abs(l.b-l.a); } ld distanceBetweenSegments(const Segment &s, const Segment &t){ if(isIntersect(s, t)) return (ld)0; return std::min({ distanceBetweenSegmentAndPoint(s, t.a), distanceBetweenSegmentAndPoint(s, t.b), distanceBetweenSegmentAndPoint(t, s.a), distanceBetweenSegmentAndPoint(t, s.b), }); } // v : 多角形の頂点、反時計回り struct Polygon{ vector v; Polygon() = default; Polygon(vector p) : v(p) {} void add(Point a) { v.eb(a); } void rem(){ assert(!v.empty()); v.pop_back(); } ld getArea(){ ld res = 0; int n = v.size(); assert(n >= 1); rep(i,n){ res += cross(v[i], v[(i+1)%n]); } return res * ld(0.5); } /** * TODO : 一直線上にある場合の判定について少し考える */ bool isConvex(){ int n = v.size(); rep(i,n){ Point pre = v[i]; Point now = v[(i+1)%n]; Point nxt = v[(i+2)%n]; if(ccw(pre, now, nxt) == -1) return false; } return true; } /** * @return * 2 : 含まれる * 1 : 辺上 * 0 : 含まれない */ int contain(Point &p){ bool in = false; int n = v.size(); rep(i,n){ Point a = v[i]-p, b = v[(i+1)%n]-p; if(imag(a) > imag(b)) { std::swap(a, b); } if(imag(a) <= EPS && EPS < imag(b) && cross(a, b) < -EPS){ in = !in; } if(equal(cross(a, b), 0) && dot(a, b) <= EPS){ return 1; } } return (in ? 2 : 0); } }; // 凸包、反時計回り // @param border // true -> border is included // false -> border is not included Polygon ConvexHull(Polygon p, bool border){ ld feps = EPS; if(border) feps = -EPS; int n = p.v.size(); if (n <= 1) return p; sort(all(p.v), [](const Point &a, const Point &b){ if(equal(real(a), real(b))) return imag(a) < imag(b); return real(a) < real(b); }); vector ret(2 * n); int k = 0; // 一直線上の3点を含める -> (< -EPS) // 含め無い -> (< EPS) rep(i,n){ while(k >= 2){ Point pre = ret[k-2], now = ret[k-1], nxt = p.v[i]; if(cross(now-pre, nxt-now) < feps) k--; else break; } ret[k++] = p.v[i]; } int t = k+1; rrep(i,n-1){ while(k >= t){ Point pre = ret[k-2], now = ret[k-1], nxt = p.v[i]; if(cross(now-pre, nxt-now) < feps) k--; else break; } ret[k++] = p.v[i]; } ret.resize(k-1); if (!border && ret.size() == 2 && equal(ret[0], ret[1])) ret.pop_back(); return Polygon(ret); } /** * @return * 4 : 2 つの円が離れている * 3 : 外接している * 2 : 2 点で交わる * 1 : 内接している * 0 : 内包している */ int isIntersect(const Circle &c1, const Circle &c2){ ld d = fabs(c1.p-c2.p); if(d > c1.r + c2.r + EPS) return 4; if(equal(d, c1.r + c2.r)) return 3; if(equal(d, fabs(c1.r-c2.r))) return 1; if(d < fabs(c1.r-c2.r)-EPS) return 0; return 2; } Circle inCircle(const Point &a, const Point &b, const Point &c){ ld A = abs(a-c), B = abs(a-c), C = abs(a-b); Point p = a * A + b * B + c * C; p /= (A+B+C); ld r = distanceBetweenLineAndPoint(Line(a,b), p); return Circle(p, r); } vector crossPoint(const Circle &c, const Line &l){ vector res; ld d = distanceBetweenLineAndPoint(l, c.p); if(d > c.r + EPS){ return res; } Point h = projection(l, c.p); if(equal(d, c.r)){ res.eb(h); return res; } Point e = unitVector(l.b-l.a); ld ph = sqrtl(c.r*c.r-d*d); res.eb(h-e*ph); res.eb(h+e*ph); return res; } vector crossPoint(const Circle &c1, const Circle &c2){ vector res; int mode = isIntersect(c1, c2); ld d = abs(c1.p-c2.p); if(mode == 4 || mode == 0){ return res; } if(mode == 3){ ld t = c1.r/(c1.r+c2.r); res.eb(c1.p+(c2.p-c1.p)*t); return res; } if(mode == 1){ if(c2.r < c1.r-EPS){ res.eb(c1.p+(c2.p-c1.p)*(c1.r/d)); } else{ res.eb(c2.p+(c1.p-c2.p)*(c2.r/d)); } return res; } // 2 円が重なる場合 ld rc1 = (c1.r * c1.r + d * d - c2.r * c2.r) / (2 * d); ld rs1 = sqrtl(c1.r * c1.r - rc1 * rc1); if(c1.r - fabs(rc1) < EPS){ rs1 = 0; } Point e12 = unitVector(c2.p-c1.p); res.eb(c1.p + rc1*e12 + rs1*normalVector(e12)); res.eb(c1.p + rc1*e12 - rs1*normalVector(e12)); return res; } vector tangentToCircle(const Point &p, const Circle &c){ if(c.r < abs(c.p-p)-EPS) return {}; if(equal(c.r, abs(c.p-p))) return {p}; return crossPoint(c, Circle(p, sqrtl(norm(c.p-p)-c.r*c.r))); } /** * TODO : 完全理解 */ vector tangent(const Circle &a, const Circle &b){ vector ret; ld g = abs(a.p-b.p); if(equal(g, 0)){ return ret; } Point u = unitVector(b.p-a.p); Point v = normalVector(u); for(int s : {-1, 1}){ ld h = (a.r + b.r * s) / g; if(equal(h*h, 1)){ ret.eb(a.p+(h>0 ? u : -u)*a.r, a.p+(h>0 ? u:-u)*a.r+v); } else if(1-h*h >= EPS){ Point U = u*h, V = v*sqrtl(1-h*h); ret.eb(a.p+(U+V)*a.r, b.p-(U+V)*(b.r*s)); ret.eb(a.p+(U-V)*a.r, b.p-(U-V)*(b.r*s)); } } return ret; } ll mostCommonPoint(vector C){ ll N = C.size(); vector cand; rep(i,N){ cand.eb(C[i].p); rep(j,i){ for(auto el: crossPoint(C[i], C[j])) cand.eb(el); } } ll ans = 0; for(auto center: cand){ ll cnt = 0; rep(i,N){ cnt += C[i].contain(center); } ans = std::max(ans, cnt); } return ans; } vector argSort(vector p) { sort(all(p), [](const Point &a, const Point &b){ auto type = [&](Point x) { if (equal(x, ORIGIN)) return 0; if (x.imag() < -EPS || (equal(x.imag(), 0) && -EPS < x.real())) return -1; return 1; }; int ta = type(a), tb = type(b); if (ta != tb) return (ta < tb); return cross(a, b) > 0; }); return p; } Point input() { ld x, y; cin >> x >> y; return Point(x, y); } pair closest_pair(vector &x, int left = 0, int right = -1){ if (right == -1) { right = x.size(); sort(all(x), [](const Point &a, const Point &b){ if (equal(a.real(), b.real())) return a.imag() < b.imag(); return a.real() < b.real(); }); } if (right-left <= 1) { return make_pair(Point(INF, INF), Point(-INF, -INF)); } int mid = (left+right)/2; auto p = x[mid]; pair res; { auto l = closest_pair(x, left, mid); auto r = closest_pair(x, mid, right); if (abs(l.first-l.second) < abs(r.first-r.second)) res = l; else res = r; } auto within = [&](ld a, ld b) { return fabs(a-b) < abs(res.first-res.second)-EPS; }; auto chminP = [&](Point a, Point b) { if (abs(a-b) < abs(res.first-res.second)) res = make_pair(a, b); }; vector y, z; { int l = left, r = mid; while(l < mid && r < right) { if (x[l].imag() < x[r].imag()) { y.eb(x[l++]); } else { y.eb(x[r++]); } } while (l < mid) y.eb(x[l++]); while (r < right) y.eb(x[r++]); rep2(i,left,right) { x[i] = y[i-left]; if (within(x[i].real(), p.real())){ z.eb(x[i]); } } } rep(i,z.size()) { rep2(j,i+1,z.size()) { if (within(z[i].imag(), z[j].imag())) chminP(z[i], z[j]); else break; } rrep2(j,i,0) { if (within(z[i].imag(), z[j].imag())) chminP(z[i], z[j]); else break; } } return res; } pair furthest_pair(vector x) { vector p = ConvexHull(Polygon(x), false).v; if(p.size() == 1) return make_pair(p[0], p[0]); if(p.size() == 2) return make_pair(p[0], p[1]); auto res = make_pair(p[0], p[1]); auto chmaxP = [&](Point a, Point b) { if (abs(a-b) > abs(res.first-res.second)) res = make_pair(a, b); }; int now = 0; rep(i,p.size()) { while(true){ chmaxP(p[i], p[now]); if(cross(p[(i+1)%p.size()]-p[i], p[(now+1)%p.size()]-p[now]) < -EPS) break; now++; now %= SZ(p); } } return res; } Point outerCenter(Point a, Point b, Point c){ ld A = norm(b-c), B = norm(c-a), C = norm(a-b); ld S = cross(c-a, b-a); return (A*(B+C-A)*a + B*(C+A-B)*b + C*(A+B-C)*c) / (ld(4)*S*S); } // 最小包含円、期待値 O(N) ld min_ball(vector p){ random_device rnd; mt19937 mt(rnd()); shuffle(all(p), mt); ll N = SZ(p); Circle c; auto upd = [&](int i, int j){ c = Circle((p[i]+p[j])/ld(2), abs(p[i]-p[j])/ld(2)); }; auto upd2 = [&](int i, int j, int k){ Point cent = outerCenter(p[i], p[j], p[k]); c = Circle(cent, abs(cent-p[i])); }; if(N == 1) return ld(0); upd(0,1); rep2(i,2,N){ if(c.contain(p[i])) continue; upd(0,i); rep(j,i){ if(c.contain(p[j])) continue; upd(j,i); rep(k,j){ if(c.contain(p[k])) continue; upd2(i,j,k); } } } return c.r; } int main(){ std::ios::sync_with_stdio(false); std::cin.tie(nullptr); std::cout.tie(nullptr); cout << std::fixed << std::setprecision(15); ll N; ld A,B; cin >> N >> A >> B; A *= 2; B *= 2; // sqrt(AB) による反転 Point c; if(N%2){ c = Point((A+B)/ld(2), ld(N/2)*(B-A)); } else{ c = Point((A+B)/ld(2), ld(N/2)*(B-A)-(B-A)/ld(2)); } Point x = c, y = c, z = c; x = Point(A, imag(c)); y = Point(B, imag(c)); z = c - Point(0, (B-A)/ld(2)); x *= A*B/norm(x); y *= A*B/norm(y); z *= A*B/norm(z); c = outerCenter(x,y,z); cout << abs(c-x) << endl; }