The two-digit integers from $19$ to $92$ are written consecutively to form the large integer $$N=192021\cdots 909192$$

Suppose that the $3^k$ is the highest power of $3$ that is a factor of $N$. What is $k$.

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered.

What is the value of \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?\]

What is the hundreds digit of $(20! - 15!)$?

Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n$?

A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn?

What is the greatest number of consecutive integers whose sum is $45$?

For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?

• a square
• a rectangle that is not a square
• a rhombus that is not a square
• a parallelogram that is not a rectangle or a rhombus
• an isosceles trapezoid that is not a parallelogram

Two lines with slopes $\frac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10$?

The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.

How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?

• some rotation around a point of line $\ell$
• some translation in the direction parallel to line $\ell$
• the reflection across line $\ell$
• some reflection across a line perpendicular to line& $\ell$

What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?

A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?

How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)?

Melanie computes the mean $\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1$s, $12$, $2$s, $\cdots$, $12$ $28$s, $11$ $29$s, $11$ $30$s, and $7$ $31$s. Let $d$ be the median of the modes. Which of the following statements is true?

$(A)$ $\mu < d < M$ $\qquad$ $(B)$ $M < d < \mu$ $\qquad$ $(C)$ $d=M=\mu$ $\qquad$ $(D)$ $d < M <\mu$ $\qquad$ $(E)$ $d < \mu < M$

Let  be an isosceles triangle with  and . Construct the circle with diameter , and let  and  be the other intersection points of the circle with the sides  and , respectively. Let  be the intersection of the diagonals of the quadrilateral . What is the degree measure of 

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 $k$?