Practice (TheColoringMethod)

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In $\triangle{ABC}$, $AB = 33, AC=21,$ and $BC=m$ where $m$ is an integer. There exist points $D$ and $E$ on $AB$ and $AC$, respectively, such that $AD=DE=EC=n$ where $n$ is also an integer. Find all the possible values of $m$.


Let quadrilateral $ABCD$ inscribe a circle. If $BE=ED$, prove $$AB^2+BC^2 +CD^2 + DA^2 = 2AC^2$$


Let $H$ be the orthocenter of acute $\triangle{ABC}$. Show that $$a\cdot BH\cdot CH + b\cdot CH\cdot AH+c\cdot AH\cdot BH=abc$$ where $a=BC, b=CA,$ and $c=AB$.

In $\triangle{ABC}$, $AE$ and $AF$ trisects $\angle{A}$, $BF$ and $BD$ trisects $\angle{B}$, $CD$ and $CE$ trisects $\angle{C}$. Show that $\triangle{DEF}$ is equilateral.


As shown, $\angle{ACB} = 90^\circ$, $AD=DB$, $DE=DC$, $EM\perp AB$, and $EN\perp CD$. Prove $$MN\cdot AB = AC\cdot CB$$


As shown, in $\triangle{ABC}$, $AB=AC$, $\angle{A} = 20^\circ$, $\angle{ABE} = 30^\circ$, and $\angle{ACD}=20^\circ$. Find the measurement of $\angle{CDE}$.


Let $O_1$ and $O_2$ be two intersecting circles. Let a common tangent to these two circles touch $O_1$ at $A$ and $O_2$ at $B$. Show that the common chord of these two circles, when extended, bisects segment $AB$.


Line $PT$ is tangent to circle $O$ at point $T$. $PA$ intersects circle $O$ and its diameter $CT$ at $B$, $D$, and $A$ in that order. If $CD=2, AD=3,$ and $BD=6$, find the length of $PB$.


In tetrahedron $ABCD$, $\angle{ADB} = \angle{BDC} = \angle{CDA} = 60^\circ$, $AD=BD=3$, and $CD=2$. Find the radius of $ABCD$'s circumsphere.


In tetrahedron $P-ABC$, $AB=BC=CA$ and $PA=PB=PC$. If $AB=1$ and the altitude from $P$ to $ABC$ is $\sqrt{2}$, find the radius of $P-ABC$'s inscribed sphere.

Let $AB=2$ is a diameter of circle $O$. If $AC=AO$, $AC\perp AB$, $BD=\frac{3}{2}\cdot AB$, $BD\perp AB$ and $P$ is a point on arc $AB$. Find the largest possible area of the enclosed polygon $ABDPC$.


Two circles, $O_1$ and $O_2$ are tangent. Let $AB$ be their common tangent line which touches $O_1$ at point $A$ and touches $O_2$ at point $B$. Extend $AO_1$ and intersects $O_1$ at another point $C$. Line $CD$ is tangent to circle $O_2$ at point $D$. Show that $AC=CD$.


Let $P$ be a point inside square $ABCD$ such that $AP=1, BP = 3,$ nd $DP=\sqrt{7}$. Find the area of $ABCD$. Try to find at least two solutions.


Let $M$ be the midpoint of $AB$ which is the hypotenuse of a non-isosceles right triangle ${ABC}$. If $DM\perp AB$ and $DC$ bisects $\angle{ACB}$, show $CM=DM$.


Let $ABCD$ be a square. Point $E$ is outside $ABCD$ such that $DE=CD$ and $\angle{AED}=15^\circ$. Show $\triangle{DCE}$ is equilateral.

Compute the value of $$\sum_{n=2019}^\infty\frac{1}{\binom{n}{2019}}$$

Show that $\sin{x}+2\sin{2x}+\cdots + n\sin{nx}=\frac{(n+1)\sin{nx} - n\sin{(n+1)x}}{2(1-\cos{x})}$

Simplify $\cos{x}\cos{2x}\cdots\cos{2^{n-1}x}$.

Evaluate $\cos\frac{2\pi}{2n+1}+\cos\frac{4\pi}{2n+1}+\cdots+\cos\frac{2n\pi}{2n+1}$.

Show that $C_n^0-C_n^2+C_n^4-C_n^6+\cdots=2^{\frac{n}{2}}\cos\frac{n\pi}{4}$.

Show that \begin{align*} C_n^0-C_n^2+C_n^4-C_n^6+\cdots &=2^{\frac{n}{2}}\cos\frac{n\pi}{4}\\ C_n^1-C_n^3+C_n^5-C_n^7+\cdots &=2^{\frac{n}{2}}\sin\frac{n\pi}{4} \end{align*}

Show that $$\binom{n}{1}-\frac{1}{2}\binom{n}{2}+\frac{1}{3}\binom{n}{3}-\cdots+(-1)^{n+1}\binom{n}{n}= 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$$


Find the value of $$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\cdots +(-1)^n\binom{n}{n}$$


John walks from point $A$ to $C$ while Mary goes from point $B$ to $D$. Both of them will move along the grid, either right or up, so they take shortest routes. How many different possibilities are there such that their routes do not intersect?


Show that $$\sum_{k=0}^n\left(2^k\binom{n}{k}\right)=3^n$$