A tree is a structure which grows from a single root node. By convention, the root node is usually placed at the top. Then, new nodes, referred as children nodes, can be attached to an existing node, referred as parent node, by edges. The root node has no parent and all other nodes have exactly one parent. A node may have any number of children nodes, including no child at all. No looping chain of nodes is permitted in this structure. For example, there are exactly $5$ different types of trees with $4$ nodes. They are shown below. The goal is to find the number of different types of tree with $n$ nodes where $n$ is a positive integer greater than $1$.

Find the generating function for the sequence $\ 0,\ 1,\ -\frac{1}{2},\ \frac{1}{3},\ -\frac{1}{4},\ \cdots$

There are $30$ identical souvenirs to be distributed among $50$ students. Each student may receive more than one souvenir. How many different distribution plans are there?

Find the number of integer solutions to the equation $a+b+c=6$ where $-1 \le a < 2$ and $1\le b,\ c\le 4$.

Let $\mathbb{A}=\{a_1,\ a_2,\ \cdots,\ a_{100}\}$ be a set containing $100$ real numbers, $\mathbb{B}=\{b_1,\ b_2,\ \cdots,\ b_{50}\}$ be a set containing $50$ real numbers, and $\mathcal{F}$ be a mapping from $\mathbb{A}$ to $\mathbb{B}$. Find the number of possible $\mathcal{F}$ if  $\mathcal{F}(a_1) \le \mathcal{F}(a_2)\le\cdots\mathcal{F}(a_1)$, and for every $b_i\in\mathbb{B}$, there exists an element $a_i\in\mathbb{A}$ such that the $\mathcal{F}(a_i)=b_i$.

Let positive integers $n$ and $k$ satisfy $n\ge 2k$. How many $k$-sided convex polygons are there whose vertices are those of an $n$-sided convex polygon and edges are diagonals of the same $n$-polygon.

Show that for any positive integer $n$, the value of $\frac{(n^2)!}{(n!)^{n+1}}$ is an integer.

Assuming positive integer $n$ satisfies $n\equiv 1\pmod{4}$ and $n > 1$. Let $\mathbb{P}=\{a_1,\ a_2,\ \cdots,\ a_n\}$ be a permutation of $\{1,\ 2,\ \cdots,\ n\}$. If $k_p$ denotes the largest index $k$ associated with $\mathbb{P}$ such that the following inequality holds $$a_1 + a_2 + \cdots + a_k < a_{k+1}+a_{k+2}+\cdots + a_n$$

Find the sum of $k_p$ for all possible $\mathbb{P}$.

How many ways are there to arrange $8$ girls and $25$ boys to sit around a table so that there are at least $2$ boys between any pair of girls? If a sitting plan can be simply rotated to match another one, these two are treated as the same.

Let $\mathbb{S}=\{1,\ 2,\ \cdots,\ 1000\}$ and $\mathbb{A}$ be a subset of $\mathbb{S}$. If the number of elements in $\mathbb{A}$ is $201$ and their sum is a multiple of $5$, then $\mathbb{A}$ is called $\textit{good}$. How many good $\mathbb{A}$ are there?

There are $n \ge 6$ points on a circle, every two points are connected by a line segment. No three diagonals are concurrent. How many triangles are created by these sides and diagonals?

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