update ucpc2026
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@@ -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
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\[a_n = d_1r_1^n + \dots + d_kr_k^n. \]
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Non-distinct roots $r$ become polynomial factors, e.g. $a_n = (d_1n + d_2)r^n$.
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% \subsection{Trigonometry}
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% \begin{align*}
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% \sin(v+w)&{}=\sin v\cos w+\cos v\sin w\\
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% \cos(v+w)&{}=\cos v\cos w-\sin v\sin w\\
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% \tan(v+w)&{}=\dfrac{\tan v+\tan w}{1-\tan v\tan w}\\
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% \sin v+\sin w&{}=2\sin\dfrac{v+w}{2}\cos\dfrac{v-w}{2}\\
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% \cos v+\cos w&{}=2\cos\dfrac{v+w}{2}\cos\dfrac{v-w}{2}\\
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% (V+W)\tan(v-w)/2&{}=(V-W)\tan(v+w)/2
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% \end{align*}
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% where $V, W$ are lengths of sides opposite angles $v, w$.
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% \begin{align*}
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% a\cos x+b\sin x&=r\cos(x-\phi)\\
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% a\sin x+b\cos x&=r\sin(x+\phi)
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% \end{align*}
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% where $r=\sqrt{a^2+b^2}, \phi=\operatorname{atan2}(b,a)$.
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\subsection{Trigonometry}
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\begin{align*}
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\sin(v+w)&{}=\sin v\cos w+\cos v\sin w\\
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\cos(v+w)&{}=\cos v\cos w-\sin v\sin w\\
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\tan(v+w)&{}=\dfrac{\tan v+\tan w}{1-\tan v\tan w}\\
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\sin v+\sin w&{}=2\sin\dfrac{v+w}{2}\cos\dfrac{v-w}{2}\\
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\cos v+\cos w&{}=2\cos\dfrac{v+w}{2}\cos\dfrac{v-w}{2}\\
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(V+W)\tan(v-w)/2&{}=(V-W)\tan(v+w)/2
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\end{align*}
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where $V, W$ are lengths of sides opposite angles $v, w$.
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\begin{align*}
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a\cos x+b\sin x&=r\cos(x-\phi)\\
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a\sin x+b\cos x&=r\sin(x+\phi)
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\end{align*}
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where $r=\sqrt{a^2+b^2}, \phi=\operatorname{atan2}(b,a)$.
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\subsection{Geometry}
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\subsubsection{Spherical coordinates}
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\begin{center}
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\includegraphics[width=25mm]{source/Math/sphericalCoordinates.pdf}
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\end{center}
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\[\begin{array}{cc}
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x = r\sin\theta\cos\phi & r = \sqrt{x^2+y^2+z^2}\\
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y = r\sin\theta\sin\phi & \theta = \textrm{acos}(z/\sqrt{x^2+y^2+z^2})\\
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z = r\cos\theta & \phi = \textrm{atan2}(y,x)
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\end{array}\]
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\subsubsection{Triangles}
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Side lengths: $a,b,c$
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Semiperimeter: $p=\dfrac{a+b+c}{2}$
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\begin{tikzpicture}[scale=1.1]
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Area: $A=\sqrt{p(p-a)(p-b)(p-c)}$
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%------------------------------------------------
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% Triangle
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%------------------------------------------------
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\tkzDefPoint(0,0){B}
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\tkzDefPoint(6,0){C}
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\tkzDefPoint(2,4.5){A}
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Circumradius: $R=\dfrac{abc}{4A}$
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Inradius: $r=\dfrac{A}{p}$
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\tkzDrawPolygon[thick](A,B,C)
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Length of the median (divides the triangle into two equal area triangles): $m_a=\tfrac{1}{2}\sqrt{2b^2+2c^2-a^2}$
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%------------------------------------------------
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% Midpoint & Median
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%------------------------------------------------
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\tkzDefMidPoint(B,C)
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\tkzGetPoint{M}
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Length of the bisector (divides angles into two): $s_a=\sqrt{bc\left[1-\left(\dfrac{a}{b+c}\right)^2\right]}$
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\tkzDrawSegment[dashed](A,M)
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\tkzLabelSegment[right](A,M){$m_a$}
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\tkzMarkSegments[mark=||, size=3pt, color=blue](B,M M,C)
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Law of sines: $\dfrac{\sin\alpha}{a}=\dfrac{\sin\beta}{b}=\dfrac{\sin\gamma}{c}=\dfrac{1}{2R}$
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Law of cosines: $a^2=b^2+c^2-2bc\cos\alpha$
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%------------------------------------------------
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% Angle bisector
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%------------------------------------------------
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\tkzDefLine[bisector](B,A,C)
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\tkzGetPoint{X}
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\tkzInterLL(A,X)(B,C)
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\tkzGetPoint{S}
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Law of tangents: $\dfrac{a+b}{a-b}=\dfrac{\tan\dfrac{\alpha+\beta}{2}}{\tan\dfrac{\alpha-\beta}{2}}$
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\tkzDrawSegment[densely dotted](A,S)
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\tkzLabelSegment[left](A,S){$s_a$}
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%------------------------------------------------
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% Incenter & Incircle
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%------------------------------------------------
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\tkzInCenter(A,B,C)
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\tkzGetPoint{I}
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\tkzDrawPoint[fill=black](I)
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\tkzDefPointBy[projection=onto B--C](I)
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\tkzGetPoint{H_a}
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\tkzDrawCircle[thick, red](I, H_a)
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\tkzDrawSegment[->](I,H_a)
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\tkzLabelSegment[left](I,H_a){$r$}
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%------------------------------------------------
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% Circumcenter & Circumcircle
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%------------------------------------------------
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\tkzCircumCenter(A,B,C)
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\tkzGetPoint{O}
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\tkzDrawCircle[thick, blue](O,A)
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\tkzDrawSegment(O,C)
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\tkzLabelSegment[above](O,C){$R$}
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%------------------------------------------------
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% Labels
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%------------------------------------------------
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\tkzLabelPoints[above](A)
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\tkzLabelPoints[left](B)
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\tkzLabelPoints[right](C)
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\tkzLabelPoints[left](I)
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\tkzLabelPoints[above](O)
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\tkzLabelPoints[below](H_a)
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\tkzLabelPoints[below](M)
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\tkzDrawPoint[fill=black](A)
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\tkzDrawPoint[fill=black](B)
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\tkzDrawPoint[fill=black](C)
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\tkzDrawPoint[fill=black](O)
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\tkzDrawPoint[fill=black](H_a)
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\tkzDrawPoint[fill=black](M)
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\tkzLabelSegment[left=0.7](A,B){$c$}
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\tkzLabelSegment[above right=0.7](A,C){$b$}
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\tkzLabelSegment[below=0.7](B,C){$a$}
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\draw[dashed] (A) to[bend right=17] (B);
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\draw[dashed] (B) to[bend right=17] (C);
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\draw[dashed] (C) to[bend right=17] (A);
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% \node at (2.7,1.7) {$A$};
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\end{tikzpicture}
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Semiperimeter $s = \dfrac{a+b+c}{2}$; Area $A=\sqrt{s(s-a)(s-b)(s-c)}$
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% Circumradius:
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% Inradius:
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% Length of the median (divides the triangle into two equal area triangles):
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$R=\dfrac{abc}{4A}$; $r=\dfrac{A}{s}$; $m_a=\tfrac{1}{2}\sqrt{2b^2+2c^2-a^2}$
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% Length of the bisector (divides angles into two):
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$s_a=\sqrt{bc\left[1-\left(\dfrac{a}{b+c}\right)^2\right]}$
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$2R=\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$
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$a=b\cos C+c\cos B$; $a^2=b^2+c^2-2bc\cos A$; $\dfrac{a+b}{a-b}=\dfrac{\tan\dfrac{\alpha+\beta}{2}}{\tan\dfrac{\alpha-\beta}{2}}$
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\subsubsection{Quadrilaterals}
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With side lengths $a,b,c,d$, diagonals $e, f$, diagonals angle $\theta$, area $A$ and
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magic flux $F=b^2+d^2-a^2-c^2$:
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\[ 4A = 2ef \cdot \sin\theta = F\tan\theta = \sqrt{4e^2f^2-F^2} \]
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\[ 4A = 2ef \sin\theta = F\tan\theta = \sqrt{4e^2f^2-F^2} \]
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For cyclic quadrilaterals the sum of opposite angles is $180^\circ$,
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$ef = ac + bd$, and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$.
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% \subsubsection{Spherical coordinates}
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% \begin{center}
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% \includegraphics[width=25mm]{source/Math/sphericalCoordinates.pdf}
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% \end{center}
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% \[\begin{array}{cc}
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% x = r\sin\theta\cos\phi & r = \sqrt{x^2+y^2+z^2}\\
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% y = r\sin\theta\sin\phi & \theta = \textrm{acos}(z/\sqrt{x^2+y^2+z^2})\\
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% z = r\cos\theta & \phi = \textrm{atan2}(y,x)
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% \end{array}\]
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\subsection{Derivatives/Integrals}
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\begin{align*}
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