For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?

A sequence of numbers is defined recursively by $a_1=1$, $a_2=\frac{3}{7}$, and $$a_n=\frac{a_{n-2}\cdot a_{n-1}}{2a_{n-2}-a_{n-1}}$$

for all $n\ge 3$. Find the value of $a_{2019}$.

The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1$?

A child builds towers using identically shaped cubes of different colors. How many different towers with a height of $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)

For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k=0.232323\cdots_k$. What is $k$?

What is the least possible value ofwhere  is a real number?

The numbers $1$, $2$, $\cdots$, $9$ are randomly placed into the $9$ squares of a $3\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?

A sphere with center $O$ has radius $6$. A triangle with sides of length $15$, $15$, and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?

Real numbers between $0$ and $1$, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is $0$ if the second flip is heads and $1$ if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,\ 1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $\mid x-y\mid > \frac{1}{2}$?

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?

Equally divide each side of a triangle into $n$ parts and then connect these points to draw lines which are parallel to one of the triangle's sides. Find the number of parallelograms created by these lines.