Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number , then Todd must say the next two numbers ( and ), then Tucker must say the next three numbers (, , ), then Tadd must say the next four numbers (, , , ), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number  is reached. What is the th number said by Tadd?

Let , , and  be the distinct roots of the polynomial . It is given that there exist real numbers , , and  such thatfor all . What is ?

For how many integers $n$ between $1$ and $50$, inclusive, is $$\frac{(n^2-1)!}{(n!)^n}$$

The area of a pizza with radius $4$ is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$?

Suppose $a$ is $150\%$ of $b$. What percent of $a$ is $3b$?

Positive real numbers $x\ne 1$ and $y\ne 1$ satisfy $\log_2x=\log_y16$ and $xy=64$. What is $\left(\log_2\frac{x}{y}\right)^2$?

How many ways are there to paint each of the integers $2$, $3$, $\cdots$, $9$ either red, green, or blue so that each number has a different color from each of its proper divisors?

For a certain complex number $c$, the polynomial $$P(x)=(x^2-2x+2)(x^2-cx+4)(x^2-4x+8)$$

has exactly $4$ distinct roots. What is $\mid c\mid$?

Positive real numbers $a$ and $b$ have the property that $$\sqrt{\log a}+\sqrt{\log b} +\log\sqrt{a} + \log\sqrt{b}=100$$

and all four terms on the left are positive integers, where $\log$ denotes the base-$10$ logarithm. What is $ab$?

Let $s_k$ denote the sum of the $k^{th}$ powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1}=as_k+bs_{k-1}+ck_{k-2}$ for $k=2$, $3$, $\cdots$. What is $a+b+c$?

In $\triangle{ABC}$ with integer side lengths, $$\cos{A}=\frac{11}{16},\qquad\cos{B}=\frac{7}{8},\qquad\text{and}\qquad\cos{C}=-\frac{1}{4}$$

What is the least possible perimeter for $\triangle{ABC}$?

Let $$z=\frac{1+i}{\sqrt{2}}$$

What is $$\left(z^{1^2}+z^{2^2}+z^{3^2}+\cdots+z^{12^2}\right)\left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\cdots+\frac{1}{z^{12^2}}\right)$$

Circles $\omega$ and $\gamma$, both centered at $O$, have radii $20$ and $17$, respectively. Equilateral triangle $ABC$, whose interior lies in the interior of $\omega$ but in the exterior of $\gamma$, has vertex $A$ on $\omega$, and the line containing side $\overline{BC}$ is tangent to $\gamma$, Segments $\overline{AO}$ and $\overline{BC}$ intersect at $P$, and $\frac{BP}{CP}=3$. Find the length of  $AB$.

Define binary operations $\diamondsuit$ and $\heartsuit$ by $$a\diamondsuit b=a^{\log_7(b)}\qquad\text{and}\qquad a\heartsuit b=a^{\frac{1}{\log_7(b)}}$$

for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3=3\heartsuit 2$ and $$a_n=(n\heartsuit (n-1))\diamondsuit a_{n-1}$$

for all integers $n\ge 4$. To the nearest integer, what is $\log_7(a_{2019})$?

Let $\triangle{A_0B_0C_0}$ be a triangle whose angle measures are exactly $59.999^{\circ}$, $60^{\circ}$, and $60.001^{\circ}$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$& to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle{A_nB_nC_n}$ is obtuse?

Alicia had two containers. The first was $\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\frac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement.

$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

In a high school wit $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?

All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?

$\textbf{(A) } (-1,2) \qquad\textbf{(B) } (0,1) \qquad\textbf{(C) } (1,-2) \qquad\textbf{(D) } (1,0) \qquad\textbf{(E) } (1,2)$

Triangle $ABC$ lies in the first quadrant. Points& $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is not always true?

$\textbf{(A) }$ Triangle $A'B'C'$" lies in the first quadrant.

$\textbf{(B) }$ Triangles $ABC$ and $A'B'C'$ have the same area.

$\textbf{(C) }$ The slope of line $AA'$ is $-1$.

$\textbf{(D) }$ The slopes of lines $AA'$ and $CC'$ are the same.

$\textbf{(E) }$ Lines $AB$ and $A'B'$ are perpendicular to each other.

There is a real $n$ such that $(n+1)! + (n+2)! = n! \cdot 440$. What is the sum of the digits of $n$?

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs $20$ cents. What is the smallest possible value of $n$?

The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length $2$ and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?

The function $\mathcal{f}$ is defined by $$f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|$$ for all real numbers $x$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number "$r$ . What is the range of $\mathcal{f}$?

In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?