Let $f(x)=x^4-4x^2+2.$ Find the smallest natural $n \in \mathbb{N}$ such that there exists $k,c \in \mathbb{N}$ with $$\left|f^k\left(\frac{n^2+1}{n}\right)-c^{144}\right| < \frac{1}{100}.$$

A $3\times2\times2$ right rectangular prism has one of its edges with length $3$ replaced with an edge of length $5$ parallel to the original edge. The other $11$ edges remain the same length, and the $6$ vertices that are not endpoints of the replaced edge remain in place. The resulting convex solid has $8$ faces, as shown below. Find the volume of the solid. [i]Proposed by Justin Hsieh[/i]

Let $ABC$ be a triangle with centroid $G$ and $BC = 3$. If $ABC$ is similar to $GAB$, compute the area of $ABC$.

On Misha's new phone, a passlock consists of six circles arranged in a $2\times 3$ rectangle. The lock is opened by a continuous path connecting the six circles; the path cannot pass through a circle on the way between two others (e.g. the top left and right circles cannot be adjacent). For example, the left path shown below is allowed but the right path is not. (Paths are considered to be oriented, so that a path starting at $A$ and ending at $B$ is different from a path starting at $B$ and ending at $A$. However, in the diagrams below, the paths are valid/invalid regardless of orientation.) How many passlocks are there consisting of all six circles? [asy] size(270); defaultpen(linewidth(0.8)); real r = 0.3, rad = 0.1, shift = 3.7; pen th = linewidth(5)+gray(0.2); for(int i=0; i<= 2;i=i+1) { for(int j=0; j<= 1;j=j+1) { fill(circle((i,j),r),gray(0.8)); fill(circle((i+shift,j),r),gray(0.8)); } draw((0,1)--(2-rad,1)^^(2,1-rad)--(2,rad)^^(2-rad,0)--(0,0),th); draw(arc((2-rad,1-rad),rad,0,90)^^arc((2-rad,rad),rad,270,360),th); draw((shift+1,0)--(shift+1,1-2*rad)^^(shift+1-rad,1-rad)--(shift+rad,1-rad)^^(shift+rad,1+rad)--(shift+2,1+rad),th); draw(arc((shift+1-rad,1-2*rad),rad,0,90)^^arc((shift+rad,1),rad,90,270),th); } [/asy]

Let $S$ be a subset of the natural numbers such that $0\in S$, and for all $n\in\mathbb N$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?

A kite with $AB = BC$ and $AD = CD$ has diagonals which satisfy $AC = 80$ and $BD = 71$. Let $AC$ and $BD$ intersect at a point $O$. Find the area of the quadrilateral formed by the circumcenters of $ABO$, $BCO$, $CDO$, and $ADO$.

Points $A$, $B$, and $C$ lie in the plane such that $AB=13$, $BC=14$, and $CA=15$. A peculiar laser is fired from $A$ perpendicular to $\overline{BC}$. After bouncing off $BC$, it travels in a direction perpendicular to $CA$. When it hits $CA$, it travels in a direction perpendicular to $AB$, and after hitting $AB$ its new direction is perpendicular to $BC$ again. If this process is continued indefinitely, the laser path will eventually approach some finite polygonal shape $T_\infty$. What is the ratio of the perimeter of $T_\infty$ to the perimeter of $\triangle ABC$?

Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$ Determine $k^2$.

Let $P$ be a point inside isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $$\angle{PAD}=\angle{PDA}=90^\circ-\angle{BPC}.$$ If $PA=14, AB=18,$ and $CD=28,$ compute the area of $ABCD.$

Amy has a six-sided die which always rolls values greater than or equal to the previous roll. She rolls the die repeatedly until she rolls a $6$. Find the expected value of the sum of all distinct values she has rolled when she finishes.

If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.

Find the integer $n$ such that \[n + \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt{\sqrt{n}}\right\rfloor = 2017.\] Here, as usual, $\lfloor\cdot\rfloor$ denotes the floor function.

For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even? [i]Proposed by Andrew Wen[/i]

We have 7 buckets labelled 0-6. Initially bucket 0 is empty, while bucket $n$ (for each $1 \leq n \leq 6$) contains the list $[1,2, \ldots, n]$. Consider the following program: choose a subset $S$ of $[1,2,\ldots,6]$ uniformly at random, and replace the contents of bucket $|S|$ with $S$. Let $\tfrac{p}{q}$ be the probability that bucket 5 still contains $[1,2, \ldots, 5]$ after two executions of this program, where $p,q$ are positive coprime integers. Find $p$.

Let $ABC$ be a triangle with $AB = AC = 10$ and $BC = 12$. Define $\ell_A$ as the line through $A$ perpendicular to $\overline{AB}$. Similarly, $\ell_B$ is the line through $B$ perpendicular to $\overline{BC}$ and $\ell_C$ is the line through $C$ perpendicular to $\overline{CA}$. These three lines $\ell_A, \ell_B, \ell_C$ form a triangle with perimeter $m/n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.

How many ordered triples $(a,b,c)$ of integers satisfy the inequality \[a^2+b^2+c^2 \leq a+b+c+2?\] Let $T = TNYWR$. David rolls a standard $T$-sided die repeatedly until he first rolls $T$, writing his rolls in order on a chalkboard. What is the probability that he is able to erase some of the numbers he's written such that all that's left on the board are the numbers $1, 2, \dots, T$ in order?

Compute the sum of all integers $x$ for which there exists an integer $y$ such that \[x^3+xy+y^3=503.\] [i]Proposed by Nathan Xiong[/i]

Evaluate \[\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.\] [i]Proposed by Nathan Xiong[/i]

Eddie assigns each of Jason, Jerry, and Jonathan a different positive integer. The three are each perfectly logical and currently know that their numbers are distinct but don't know each other's numbers. Additionally, if one of them knows the answer to the question they will say so immediately. They have the following conversation listed below in chronological order: [list] [*] Eddie: Does anyone know who has the smallest number? [*] Jason, Jerry, Jonathan (at the same time): I'm not sure. [*] Jonathan: Now I know who has the smallest number. [*] Eddie: Does anyone know who has the largest number? [*] Jason, Jonathan, Jerry (at the same time): I'm not sure. [*] Jerry: Now I know who has the largest number. [*] Jason: Wow, our numbers are in an geometric sequence! [/list] Find the sum of their numbers.

Suppose $a_1, a_2, a_3, \dots,$ is a sequence of real numbers such that $$a_n = \frac{a_{n-1}a_{n-2}}{3a_{n-2}-2a_{n-1}}$$ for all $n \ge 3$. If $a_1 = 1$ and $a_{10} = 10$, what is $a_{19}$? [i]Proposed by Howard Halim[/i]

Define $\{p_n\}_{n=0}^\infty\subset\mathbb N$ and $\{q_n\}_{n=0}^\infty\subset\mathbb N$ to be sequences of natural numbers as follows: [list] [*]$p_0=q_0=1$; [*]For all $n\in\mathbb N$, $q_n$ is the smallest natural number such that there exists a natural number $p_n$ with $\gcd(p_n,q_n)=1$ satisfying \[\dfrac{p_{n-1}}{q_{n-1}} < \dfrac{p_n}{q_n} < \sqrt 2.\] [/list] Find $q_3$.

Compute the value of $$\sin^2\left(\frac{\pi}{7}\right) + \sin^2\left(\frac{3\pi}{7}\right) + \sin^2\left(\frac{5\pi}{7}\right).$$ Your answer should not involve any trigonometric functions. [i]Proposed by Howard Halim[/i]

Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.

We say that a binary string $s$ [i]contains[/i] another binary string $t$ if there exist indices $i_1,i_2,\ldots,i_{|t|}$ with $i_1 < i_2 < \ldots < i_{|t|}$ such that $$s_{i_1}s_{i_2}\ldots s_{i_{|t|}} = t.$$ (In other words, $t$ is found as a not necessarily contiguous substring of $s$.) For example, $110010$ contains $111$. What is the length of the shortest string $s$ which contains the binary representations of all the positive integers less than or equal to $2048$?

Over all natural numbers $n$ with 16 (not necessarily distinct) prime divisors, one of them maximizes the value of $s(n)/n$, where $s(n)$ denotes the sum of the divisors of $n$. What is the value of $d(d(n))$, where $d(n)$ is the the number of divisors of $n$?