update ucpc2026

2026-07-11 09:38:32 +09:00
parent 7176febe54
commit 0b8c2864ae
20 changed files with 329 additions and 163 deletions
@@ -1,32 +1,26 @@
 namespace GMS {
 template<ll mod>
 ll pow(ll a, ll b) {
-    a %= mod;
-    ll ret = 1;
+    ll ret = 1; a %= mod;
    while(b != 0) {
        if(b&1) ret = ret*a%mod;
-        a = a*a%mod; b>>=1;
+        a = a*a%mod, b/=2;
    }
    return ret;
 }
 template<ll mod, ll w>
-void ntt(vector<ll> &a, bool inv = false) {
+void ntt(vl& a, bool inv = false) {
    static_assert(mod <= (ll)2e9, "mod should be less than 2e9");
    int n = a.size(), j = 0;
-    
    assert((n & -n) == n && (mod-1)%n == 0);

    for(int i=1; i<n; i++) {
-        int bit = (n >> 1);
-        while(j >= bit) {
-            j -= bit;
-            bit >>= 1;
-        }
-        j += bit;
-        if(i < j) swap(a[i], a[j]);
+        int bit = n/2;
+        while(j >= bit) {j -= bit; bit >>= 1;}
+        j += bit; if(i < j) swap(a[i], a[j]);
    }
    
-    static vector<ll> root[30], iroot[30];
+    static vl root[30], iroot[30];
    for(int st=1; (1<<st) <= n; st++) {
        if(root[st].empty()) {
            ll t = pow<mod>(w, (mod-1)/(1<<st));
@@ -46,7 +40,6 @@ void ntt(vector<ll> &a, bool inv = false) {
    }
    
    vector<ll>* r = (inv?root:iroot);
-    
    for(int st = 1; (1<<st) <= n; st++) {
        int i = 1<<st; //int step = n / i;
        for(int j=0; j<n; j+=i) {
@@ -70,12 +63,9 @@ vl conv(vl A, vl B) {
    A.resize(t); B.resize(t);

    ntt<mod, w>(A); ntt<mod, w>(B);
-
    fors(i, 0, t-1) A[i] = A[i]*B[i]%mod;
-    
    ntt<mod, w>(A, true);
-    A.resize(n+m-1);
    
-    return A;
+    return A.resize(n+m-1), A;
 }
 } // namespace GMS
@@ -51,7 +51,7 @@ struct Qring : public vl {
    }
    friend poly operator*(const poly& A, const poly& B) {
        poly ret = conv<mod, w>(A, B);
-        // ACL : poly ret = atcoder::convolution<mod>(A, B);
+        // poly ret = atcoder::convolution<mod>(A, B); // ACL
        return ret.adjust(), ret;
    }
    friend poly inv(const poly& A, int t) { assert(A[0] != 0);
@@ -89,16 +89,15 @@ struct Qring : public vl {
        ll idx = 0; while(ret[idx] == 0) idx++;
        if((__int128_t) idx * b >= t) return poly(0, t);
        
-        ll c = ret[idx]; ll ic = pow<mod>(ret[idx], mod-2); poly g;
-        int n = ret.size();
+        ll c = ret[idx]; ll ic = pow<mod>(ret[idx], mod-2);
+        poly g; int n = ret.size();
        fors(i, idx, n-1) g[i-idx] = ret[i]*ic%mod;
        g.resize(t-idx*b);
-        
        g = exp(b * log(g, t-idx*b), t-idx*b);
        c = pow<mod>(c, b);
        
-        ret = poly(0, t); fors(i, idx*b, t-1) ret[i] = g[i-idx*b] * c % mod;
-        
+        ret = poly(0, t);
+        fors(i, idx*b, t-1) ret[i] = g[i-idx*b] * c % mod;
        return ret;
    }
    
@@ -1,16 +1,14 @@
+random_device rd; mt19937 rng(rd());
+
 ll primary_root(ll p) {
-    std::random_device rd;
-    std::mt19937 gen(rd());
-    std::uniform_int_distribution<ll> distrib(1, p-1);
-    
-    //distrib(gen);
+    uniform_int_distribution<ll> dist(1, p-1);
+    auto gen = bind(dist, rng);
+
    vl g = po_rho(p-1);
-    
    while(true) {
-        ll c = distrib(gen);
-        
-        bool ok = true; ll u = p-1;
-        ll b = 1;
+        ll c = gen(), u = p-1, b = 1;
+        bool ok = true;
+
        for(auto i:g) {
            if(i != b) u = p-1;
            ll x = pow(c, u/i, p);
@@ -33,64 +33,154 @@ If $a_n = c_1 a_{n-1} + \dots + c_k a_{n-k}$, and $r_1, \dots, r_k$ are distinct
 \[a_n = d_1r_1^n + \dots + d_kr_k^n. \]
 Non-distinct roots $r$ become polynomial factors, e.g. $a_n = (d_1n + d_2)r^n$.

-% \subsection{Trigonometry}
-% \begin{align*}
-% \sin(v+w)&{}=\sin v\cos w+\cos v\sin w\\
-% \cos(v+w)&{}=\cos v\cos w-\sin v\sin w\\
-% \tan(v+w)&{}=\dfrac{\tan v+\tan w}{1-\tan v\tan w}\\
-% \sin v+\sin w&{}=2\sin\dfrac{v+w}{2}\cos\dfrac{v-w}{2}\\
-% \cos v+\cos w&{}=2\cos\dfrac{v+w}{2}\cos\dfrac{v-w}{2}\\
-% (V+W)\tan(v-w)/2&{}=(V-W)\tan(v+w)/2
-% \end{align*}
-% where $V, W$ are lengths of sides opposite angles $v, w$.
-% \begin{align*}
-% 	a\cos x+b\sin x&=r\cos(x-\phi)\\
-% 	a\sin x+b\cos x&=r\sin(x+\phi)
-% \end{align*}
-% where $r=\sqrt{a^2+b^2}, \phi=\operatorname{atan2}(b,a)$.
+\subsection{Trigonometry}
+\begin{align*}
+\sin(v+w)&{}=\sin v\cos w+\cos v\sin w\\
+\cos(v+w)&{}=\cos v\cos w-\sin v\sin w\\
+\tan(v+w)&{}=\dfrac{\tan v+\tan w}{1-\tan v\tan w}\\
+\sin v+\sin w&{}=2\sin\dfrac{v+w}{2}\cos\dfrac{v-w}{2}\\
+\cos v+\cos w&{}=2\cos\dfrac{v+w}{2}\cos\dfrac{v-w}{2}\\
+(V+W)\tan(v-w)/2&{}=(V-W)\tan(v+w)/2
+\end{align*}
+where $V, W$ are lengths of sides opposite angles $v, w$.
+\begin{align*}
+	a\cos x+b\sin x&=r\cos(x-\phi)\\
+	a\sin x+b\cos x&=r\sin(x+\phi)
+\end{align*}
+where $r=\sqrt{a^2+b^2}, \phi=\operatorname{atan2}(b,a)$.

 \subsection{Geometry}

+\subsubsection{Spherical coordinates}
+\begin{center}
+\includegraphics[width=25mm]{source/Math/sphericalCoordinates.pdf}
+\end{center}
+\[\begin{array}{cc}
+x = r\sin\theta\cos\phi & r = \sqrt{x^2+y^2+z^2}\\
+y = r\sin\theta\sin\phi & \theta = \textrm{acos}(z/\sqrt{x^2+y^2+z^2})\\
+z = r\cos\theta & \phi = \textrm{atan2}(y,x)
+\end{array}\]
+
 \subsubsection{Triangles}
-Side lengths: $a,b,c$

-Semiperimeter: $p=\dfrac{a+b+c}{2}$
+\begin{tikzpicture}[scale=1.1]

-Area: $A=\sqrt{p(p-a)(p-b)(p-c)}$
+%------------------------------------------------
+% Triangle
+%------------------------------------------------
+\tkzDefPoint(0,0){B}
+\tkzDefPoint(6,0){C}
+\tkzDefPoint(2,4.5){A}

-Circumradius: $R=\dfrac{abc}{4A}$

-Inradius: $r=\dfrac{A}{p}$
+\tkzDrawPolygon[thick](A,B,C)

-Length of the median (divides the triangle into two equal area triangles): $m_a=\tfrac{1}{2}\sqrt{2b^2+2c^2-a^2}$
+%------------------------------------------------
+% Midpoint & Median
+%------------------------------------------------
+\tkzDefMidPoint(B,C)
+\tkzGetPoint{M}

-Length of the bisector (divides angles into two): $s_a=\sqrt{bc\left[1-\left(\dfrac{a}{b+c}\right)^2\right]}$
+\tkzDrawSegment[dashed](A,M)
+\tkzLabelSegment[right](A,M){$m_a$}
+\tkzMarkSegments[mark=||, size=3pt, color=blue](B,M M,C)

-Law of sines: $\dfrac{\sin\alpha}{a}=\dfrac{\sin\beta}{b}=\dfrac{\sin\gamma}{c}=\dfrac{1}{2R}$

-Law of cosines: $a^2=b^2+c^2-2bc\cos\alpha$
+%------------------------------------------------
+% Angle bisector
+%------------------------------------------------
+\tkzDefLine[bisector](B,A,C)
+\tkzGetPoint{X}
+\tkzInterLL(A,X)(B,C)
+\tkzGetPoint{S}

-Law of tangents: $\dfrac{a+b}{a-b}=\dfrac{\tan\dfrac{\alpha+\beta}{2}}{\tan\dfrac{\alpha-\beta}{2}}$
+\tkzDrawSegment[densely dotted](A,S)
+\tkzLabelSegment[left](A,S){$s_a$}
+
+%------------------------------------------------
+% Incenter & Incircle
+%------------------------------------------------
+\tkzInCenter(A,B,C)
+\tkzGetPoint{I}
+\tkzDrawPoint[fill=black](I)
+
+\tkzDefPointBy[projection=onto B--C](I)
+\tkzGetPoint{H_a}
+
+
+\tkzDrawCircle[thick, red](I, H_a)
+
+\tkzDrawSegment[->](I,H_a)
+\tkzLabelSegment[left](I,H_a){$r$}
+
+%------------------------------------------------
+% Circumcenter & Circumcircle
+%------------------------------------------------
+\tkzCircumCenter(A,B,C)
+\tkzGetPoint{O}
+
+\tkzDrawCircle[thick, blue](O,A)
+
+\tkzDrawSegment(O,C)
+\tkzLabelSegment[above](O,C){$R$}
+
+%------------------------------------------------
+% Labels
+%------------------------------------------------
+\tkzLabelPoints[above](A)
+\tkzLabelPoints[left](B)
+\tkzLabelPoints[right](C)
+\tkzLabelPoints[left](I)
+\tkzLabelPoints[above](O)
+\tkzLabelPoints[below](H_a)
+\tkzLabelPoints[below](M)
+
+\tkzDrawPoint[fill=black](A)
+\tkzDrawPoint[fill=black](B)
+\tkzDrawPoint[fill=black](C)
+\tkzDrawPoint[fill=black](O)
+\tkzDrawPoint[fill=black](H_a)
+\tkzDrawPoint[fill=black](M)
+
+
+\tkzLabelSegment[left=0.7](A,B){$c$}
+\tkzLabelSegment[above right=0.7](A,C){$b$}
+\tkzLabelSegment[below=0.7](B,C){$a$}
+
+\draw[dashed] (A) to[bend right=17] (B);
+\draw[dashed] (B) to[bend right=17] (C);
+\draw[dashed] (C) to[bend right=17] (A);
+
+% \node at (2.7,1.7) {$A$};
+
+\end{tikzpicture}
+
+
+Semiperimeter $s = \dfrac{a+b+c}{2}$; Area $A=\sqrt{s(s-a)(s-b)(s-c)}$
+
+% Circumradius: 
+% Inradius: 
+% Length of the median (divides the triangle into two equal area triangles): 
+$R=\dfrac{abc}{4A}$; $r=\dfrac{A}{s}$; $m_a=\tfrac{1}{2}\sqrt{2b^2+2c^2-a^2}$
+
+
+% Length of the bisector (divides angles into two): 
+$s_a=\sqrt{bc\left[1-\left(\dfrac{a}{b+c}\right)^2\right]}$
+
+$2R=\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$
+
+$a=b\cos C+c\cos B$; $a^2=b^2+c^2-2bc\cos A$; $\dfrac{a+b}{a-b}=\dfrac{\tan\dfrac{\alpha+\beta}{2}}{\tan\dfrac{\alpha-\beta}{2}}$


 \subsubsection{Quadrilaterals}
 With side lengths $a,b,c,d$, diagonals $e, f$, diagonals angle $\theta$, area $A$ and
 magic flux $F=b^2+d^2-a^2-c^2$:

-\[ 4A = 2ef \cdot \sin\theta = F\tan\theta = \sqrt{4e^2f^2-F^2} \]
+\[ 4A = 2ef \sin\theta = F\tan\theta = \sqrt{4e^2f^2-F^2} \]

 For cyclic quadrilaterals the sum of opposite angles is $180^\circ$,
 $ef = ac + bd$, and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$.

-% \subsubsection{Spherical coordinates}
-% \begin{center}
-% \includegraphics[width=25mm]{source/Math/sphericalCoordinates.pdf}
-% \end{center}
-% \[\begin{array}{cc}
-% x = r\sin\theta\cos\phi & r = \sqrt{x^2+y^2+z^2}\\
-% y = r\sin\theta\sin\phi & \theta = \textrm{acos}(z/\sqrt{x^2+y^2+z^2})\\
-% z = r\cos\theta & \phi = \textrm{atan2}(y,x)
-% \end{array}\]

 \subsection{Derivatives/Integrals}
 \begin{align*}