Practice (EndingDigits,TheDivideByNineMethod,MODBasic)

3136

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 $m$.

3137

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

3138

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 $c=AB$.

3139

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.

3140

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

3141

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

3142

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 $AB$.

3143

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 $PB$.

3144

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

3145

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.

3146

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 $ABDPC$.

3147

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 $AC=CD$.

3148

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.

3149

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 $CM=DM$.

3150

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

3151

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

3152

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

3153

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

3154

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

3155

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

3156

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*}

3157

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

3158

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

3159

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?

3160

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

Practice (EndingDigits,TheDivideByNineMethod,MODBasic)

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