The equation $ x^3 \minus{} 9x^2 \plus{} 8x \plus{} 2 \equal{} 0$ has three real roots $ p$, $ q$, $ r$. Find $ \frac {1}{p^2} \plus{} \frac {1}{q^2} \plus{} \frac {1}{r^2}$.

$ x$ and $ y$ are real numbers. $A)$ If it is known that $ x \plus{} y$ and $ x \plus{} y^2$ are rational numbers, can we conclude that $ x$ and $ y$ are also rational numbers. $B)$ If it is known that $ x \plus{} y$ , $ x \plus{} y^2$ and $ x \plus{} y^3$ are rational numbers, can we conclude that $ x$ and $ y$ are also rational numbers.

[u]Round 1[/u] [b]p1.[/b] How many ways are there to arrange the letters in the word $NEVERLAND$ such that the $2$ $N$’s are adjacent and the two $E$’s are adjacent? Assume that letters that appear the same are not distinct. [b]p2.[/b] In rectangle $ABCD$, $E$ and $F$ are on $AB$ and $CD$, respectively such that $DE = EF = FB$ and $\angle CDE = 45^o$. Find $AB + AD$ given that $AB$ and $AD$ are relatively prime positive integers. [b]p3.[/b] Maisy Airlines sees $n$ takeoffs per day. Find the minimum value of $n$ such that theremust exist two planes that take off within aminute of each other. [u]Round 2[/u] [b]p4.[/b] Nick is mixing two solutions. He has $100$ mL of a solution that is $30\%$ $X$ and $400$ mL of a solution that is $10\%$ $X$. If he combines the two, what percent $X$ is the final solution? [b]p5.[/b] Find the number of ordered pairs $(a,b)$, where $a$ and $b$ are positive integers, such that $$\frac{1}{a}+\frac{2}{b}=\frac{1}{12}.$$ [b]p6.[/b] $25$ balls are arranged in a $5$ by $5$ square. Four of the balls are randomly removed from the square. Given that the probability that the square can be rotated $180^o$ and still maintain the same configuration can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime, find $m+n$. [u]Round 3[/u] [b]p7.[/b] Maisy the ant is on corner $A$ of a $13\times 13\times 13$ box. She needs to get to the opposite corner called $B$. Maisy can only walk along the surface of the cube and takes the path that covers the least distance. Let $C$ and $D$ be the possible points where she turns on her path. Find $AC^2 + AD^2 +BC^2 +BD^2 - AB^2 -CD^2$. [b]p8.[/b] Maisyton has recently built $5$ intersections. Some intersections will get a park and some of those that get a park will also get a chess school. Find how many different ways this can happen. [b]p9.[/b] Let $f (x) = 2x -1$. Find the value of $x$ that minimizes $| f ( f ( f ( f ( f (x)))))-2020|$. [u]Round 4[/u] [b]p10.[/b] Triangle $ABC$ is isosceles, with $AB = BC > AC$. Let the angle bisector of $\angle A$ intersect side $\overline{BC}$ at point $D$, and let the altitude from $A$ intersect side $\overline{BC}$ at point $E$. If $\angle A = \angle C= x^o$, then the measure of $\angle DAE$ can be expressed as $(ax -b)^o$, for some constants $a$ and $b$. Find $ab$. [b]p11[/b]. Maisy randomly chooses $4$ integers $w$, $x$, $y$, and $z$, where $w, x, y, z \in \{1,2,3, ... ,2019,2020\}$. Given that the probability that $w^2 + x^2 + y^2 + z^2$ is not divisible by $4$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p12.[/b] Evaluate $$-\log_4 \left(\log_2 \left(\sqrt{\sqrt{\sqrt{...\sqrt{16}}}} \right)\right),$$ where there are $100$ square root signs. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

$\lim \limits_{x\to \infty} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^2}}=$ [list=1] [*] $\sqrt{e}$ [*] $\infty$ [*] Does not exists [*] None of these [/list]

Let $n \ge 2$ and $k \ge1$ be positive integers. In a country there are $n$ cities and between each pair of cities there is a bus connection in both directions. Let $A$ and $B$ be two different cities. Prove that the number of ways in which you can travel from $A$ to $B$ by using exactly $k$ buses is equal to $\frac{(n - 1)^k - (-1)^k}{n}$ .

Let $f(n)$ be the smallest prime which divides $n^4+1$. What is the remainder when the sum $f(1)+f(2)+\cdots+f(2014)$ is divided by $8$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $AB$ be a diameter of circle $\omega$. $\ell$ is the tangent line to $\omega$ at $B$. Take two points $C$, $D$ on $\ell$ such that $B$ is between $C$ and $D$. $E$, $F$ are the intersections of $\omega$ and $AC$, $AD$, respectively, and $G$, $H$ are the intersections of $\omega$ and $CF$, $DE$, respectively. Prove that $AH=AG$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the equation \[|x|f(y)+yf(x)=f(xy)+f(x^2)+f(f(y))\] for all real numbers $x$ and $y$.

A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$. A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that C lies between the lines $c = AB$ and $C'$. Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$. Prove that the three lines, joining the midpoints of pairs of parallel sides of the two triangles, have a common point.

Al travels for $20$ miles per hour rolling down a hill in his chair for two hours, then four miles per hour climbing a hill for six hours. What is his average speed, in miles per hour?

Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$. (Edited)

Let $ABCD$ be a parallelogram whose diagonals meet at $P$. Denote by $M$ the midpoint of $AB$. Let $Q$ be a point such that $QA$ is tangent to the circumcircle of $MAD$ and $QB$ is tangent to the circumcircle of $MBC$. Prove that points $Q,M,P$ are collinear. (Patrik Bak, Slovakia)

Let $n>1$ be a positive integer. Find the number of binary strings $(a_1, a_2, \ldots, a_n)$, such that the number of indices $1\leq i \leq n-1$ such that $a_i=a_{i+1}=0$ is equal to the number of indices $1 \leq i \leq n-1$, such that $a_i=a_{i+1}=1$.

Let $\vartriangle ABC$ be a triangle with $G$ as its centroid, which is the intersection of the three medians of the triangle, as shown in the diagram. If $\overline{GA} \perp \overline{GB}$ and $AB = 7$, compute $AC^2 + BC^2$. [img]https://cdn.artofproblemsolving.com/attachments/e/1/240be132c6adcfde0334a000e1f916a6292907.png[/img]

Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?

Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.

Find the roots of the equation $(x-a)(x-b)=(x-c)(x-d)$, if you know that $a+d=b+c=2015$ and $a \ne c$ (numbers $a, b, c, d$ are not given).

Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

There are seven rays emanating from a point $A$ on a plane, such that the angle between the two consecutive rays is $30 ^{\circ}$. A point $A_1$ is located on the first ray. The projection of $A_1$ onto the second ray is denoted as $A_2$. Similarly, the projection of $A_2$ onto the third ray is $A_3$, and this process continues until the projection of $A_6$ onto the seventh ray is $A_7$. Find the ratio $\frac{A_7A}{A_1A}$. [img]https://i.imgur.com/oxixe5q.png[/img] [i]Proposed by Giorgi Arabidze, Georgia[/i]

Consider functions $f$ satisfying the following four conditions: (1) $f$ is real-valued and defined for all real numbers. (2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$. (3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$. (4) We have $f(2)=4$. Prove that: a) There is a function $f$ with $f(3)=9$ satisfying the four conditions. b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$.

Let $ABC$ be an equilateral triangle. Let $Q$ be a random point on $BC$, and let $P$ be the meeting point of $AQ$ and the circumscribed circle of $\triangle ABC$. Prove that $\frac{1}{PQ}=\frac{1}{PB}+\frac{1}{PC}$.

(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$. The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$. Prove that $BU = BA$ if, and only if, $CP = CA$. (ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$. The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$. Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$. Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.

Define the function $f:\mathbb{R} \backslash \{-1,1\} \to \mathbb{R}$ to be \[f(x) = \sum_{a,b=0}^{\infty} \frac{x^{2^a3^b}}{1-x^{2^{a+1}3^{b+1}}} .\] Suppose that $f\left(y\right)-f\left(\tfrac{1}{y}\right)=2016$. Then $y$ can be written in simplest form as $\tfrac{p}{q}$. Find $p+q$. ($\mathbb{R} \backslash \{-1,1\}$ refers to the set of real numbers excluding $-1$ and $1$.)

Let $t_1 < t_2 < t_3 < \cdots < t_{99}$ be real numbers. Consider a function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x)=|x-t_1|+|x-t_2|+...+|x-t_{99}|$ . Show that $f(x)$ will attain minimum value at $x=t_{50}$.