Let $ABC$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $PD,PE,PF$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen, \[ \frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}. \]

[b]p29.[/b] Consider pentagon $ABCDE$. How many paths are there from vertex $A$ to vertex $E$ where no edge is repeated and does not go through $E$. [b]p30.[/b] Let $a_1, a_2, ...$ be a sequence of positive real numbers such that $\sum^{\infty}_{n=1} a_n = 4$. Compute the maximum possible value of $\sum^{\infty}_{n=1}\frac{\sqrt{a_n}}{2^n}$ (assume this always converges). [b]p31.[/b] Define function $f(x) = x^4 + 4$. Let $$P =\prod^{2021}_{k=1} \frac{f(4k - 1)}{f(4k - 3)}.$$ Find the remainder when $P$ is divided by $1000$. [b]p32.[/b] Reduce the following expression to a simplified rational: $\cos^7 \frac{\pi}{9} + \cos^7 \frac{5\pi}{9}+ \cos^7 \frac{7\pi}{9}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\alpha$ be a real number with $\alpha>1$. Let the sequence $(a_n)$ be defined as $$a_n=1+\sqrt[\alpha]{2+\sqrt[\alpha]{3+\ldots+\sqrt[\alpha]{n+\sqrt[\alpha]{n+1}}}}$$ for all positive integers $n$. Show that there exists a positive real constant $C$ such that $a_n<C$ for all positive integers $n$.

Let $n$ be an odd integer and $m=\phi(n)$ be the Euler's totient function. Call a set of residues $T=\{a_1, \cdots, a_k\} \pmod n$ to be [i]good[/i] if $\gcd(a_i, n) > 1$ $\forall i$, and $\gcd(a_i, a_j) = 1, \forall i \neq j$. Define the set $S_n$ consisting of the residues $$\sum_{i=1}^k a_i ^m\pmod{n}$$ over all possible residue sets $T=\{a_1,\cdots,a_k\}$ that is good. Determine $|S_n|$. [i]Proposed by Anzo Teh Zhao Yang[/i]

Determine the greatest common divisor of the numbers: $$5^5-5, 7^7-7, 9^9-9 ,..., 2017^{2017}-2017,$$

A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one?

The product $$\left(\frac{1}{2^3-1}+\frac12\right)\left(\frac{1}{3^3-1}+\frac12\right)\left(\frac{1}{4^3-1}+\frac12\right)\cdots\left(\frac{1}{100^3-1}+\frac12\right)$$ can be written as $\frac{r}{s2^t}$ where $r$, $s$, and $t$ are positive integers and $r$ and $s$ are odd and relatively prime. Find $r+s+t$.

In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\overline{AD}$. What is the area of $\triangle AMC$? $\textbf{(A) }12\qquad\textbf{(B) }15\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad \textbf{(E) }24$

Determine all real numbers $x, y, z \in (0, 1)$ that satisfy simultaneously the conditions: $(x^2 + y^2)\sqrt{1- z^2}\ge z$ $(y^2 + z^2)\sqrt{1- x^2}\ge x$ $(z^2 + x^2)\sqrt{1- y^2}\ge y$

Which is greater: $ 17091982!^2$ or $ 17091982^{17091982}$?

$C$ is the circle center $(0,1)$, radius $1$. $P$ is the parabola $y = ax^2$. They meet at $(0, 0)$. For what values of $a$ do they meet at another point or points?

In $\triangle ABC$, if $A,B,C$ are geometric series, and $b^2-a^2=ac$, then $B=$________.

Consider all pairs of points $(a,b,c)$ and $(d,e, f )$ in the $3$-D coordinate system with $ad +be +c f = -2023$. What is the least positive integer that can be the distance between such a pair of points? [i]Proposed by William Hua[/i]

Let $\alpha$ and $\beta$ be the two roots of $x^2+2017x+k$. What is the sum of the possible values of $k$ so that the lines \begin{align*} y&=2\alpha x+2017^2\\ y&=3\alpha x+2017^3 \end{align*} are perpendicular?

Given a triangle $XYZ$ with $\angle Y = 90^{\circ}, XY=1,$ and $XZ=2,$ mark a point $Q$ on $YZ$ such that $\frac{ZQ}{ZY}=\frac{1}{3}.$ A laser beam is shot from $Q$ perpendicular to $YZ,$ and it reflects off the sides of $XYZ$ indefinitely. How many bounces does it take for the laser beam to get back to $Q$ for the first time (not including the release from $Q$ and the return to $Q$)?

Let $a=\lfloor \sqrt{n} \rfloor$ for given positive integer $n.$ Express the summation $\sum_{k=1}^{n}\lfloor \sqrt{k} \rfloor$ in terms of $n$ and $a.$

There are five dots arranged in a line from left to right. Each of the dots is colored from one of five colors so that no $3$ consecutive dots are all the same color. How many ways are there to color the dots?

Let $r_1,r_2,r$, with $r_1 < r_2 < r$, be the radii of three circles $\Gamma_1,\Gamma_2,\Gamma$, respectively. The circles $\Gamma_1,\Gamma_2$ are internally tangent to $\Gamma$ at two distinct points $A,B$ and intersect in two distinct points. Prove that the segment $AB$ contains an intersection point of $\Gamma_1$ and $\Gamma_2$ if and only if $r_1 +r_2 = r$.

Prove that there are exactly $2(2^{n-1}-1)$ ways of dealing $n$ cards to two persons. (The persons may receive unequal numbers of cards.)

Find a pair prime numbers $(p, q)$, $p> q$ of , if any, such that $\frac{p^2 - q^2}{4}$ is an odd integer.

One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing $4.5$ ounces was $ \text{(A)}\ \text{96 cents}\qquad\text{(B)}\ \text{1.07 dollars}\qquad\text{(C)}\ \text{1.18 dollars}\qquad\text{(D)}\ \text{1.20 dollars}\qquad\text{(E)}\ \text{1.40 dollars} $

Let $\Gamma$ be a circle with chord $AB$ such that the length of $AB$ is greater than the radius, $r$, of $\Gamma$. Let $C$ be the point on the chord $AB$ satisfying $AC = r$. The perpendicular bisector of $BC$ intersects $\Gamma$ in the points $D$ and $E$. Lines $DC$ and $EC$ intersect $\Gamma$ for a second time at points $F$ and $G$, respectively. Given that $CD=2$ and $CE=3$, find $GF^2$. [i]Team #9[/i]

In the right triangle $ABC$ shown, $E$ and $D$ are the trisection points of the hypotenuse $AB$. If $CD=7$ and $CE=6$, what is the length of the hypotenuse $AB$? Express your answer in simplest radical form. [asy] pair A, B, C, D, E; A=(0,2.9); B=(2.1,0); C=origin; D=2/3*A+1/3*B; E=1/3*A+2/3*B; draw(A--B--C--cycle); draw(C--D); draw(C--E); label("$A$",A,N); label("$B$",B,SE); label("$C$",C,SW); label("$D$",D,NE); label("$E$",E,NE); [/asy]

Let $P$ and $Q$ be the midpoints of non-parallel chords $k_1$ and $k_2$ of a circle $\omega$, respectively. Let the tangent lines of $\omega$ passing through the endpoints of $k_1$ intersect at $A$ and the tangent lines passing through the endpoints of $k_2$ intersect at $B$. Let the symmetric point of the orthocenter of triangle $ABP$ with respect to the line $AB$ be $R$ and let the feet of the perpendiculars from $R$ to the lines $AP, BP, AQ, BQ$ be $R_1, R_2, R_3, R_4$, respectively. Prove that \[ \frac{AR_1}{PR_1} \cdot \frac{PR_2}{BR_2} = \frac{AR_3}{QR_3} \cdot \frac{QR_4}{BR_4} \]

Call a polygon [i]normal[/i] if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?