Let $N$ be the number of sequences of natural numbers $d_1,d_2,\dots,d_{10}$ such that the following conditions hold: $d_1|d_2$, $\dots$, $d_9|d_{10}$ and $d_{10}|6^{2018}$. Evaluate the remainder when $N$ is divided by $2017$.

Three friends have a total of $6$ identical pencils, and each one has at least one pencil. In how many ways can this happen? $\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$

Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $ 32$? $ \textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad \textbf{(E)}\ 2255$

[b]p1.[/b] The following number is the product of the divisors of $n$. $$2^63^3$$ What is $n$? [b]p2.[/b] Let a right triangle have the sides $AB =\sqrt3$, $BC =\sqrt2$, and $CA = 1$. Let $D$ be a point such that $AD = BD = 1$. Let $E$ be the point on line $BD$ that is equidistant from $D$ and $A$. Find the angle $\angle AEB$. [b]p3.[/b] There are twelve indistinguishable blackboards that are distributed to eight different schools. There must be at least one board for each school. How many ways are there of distributing the boards? [b]p4.[/b] A Nishop is a chess piece that moves like a knight on its first turn, like a bishop on its second turn, and in general like a knight on odd-numbered turns and like a bishop on even-numbered turns. A Nishop starts in the bottom-left square of a $3\times 3$-chessboard. How many ways can it travel to touch each square of the chessboard exactly once? [b]p5.[/b] Let a Fibonacci Spiral be a spiral constructed by the addition of quarter-circles of radius $n$, where each $n$ is a term of the Fibonacci series: $$1, 1, 2, 3, 5, 8,...$$ (Each term in this series is the sum of the two terms that precede it.) What is the arclength of the maximum Fibonacci spiral that can be enclosed in a rectangle of area $714$, whose side lengths are terms in the Fibonacci series? [b]p6.[/b] Suppose that $a_1 = 1$ and $$a_{n+1} = a_n -\frac{2}{n + 2}+\frac{4}{n + 1}-\frac{2}{n}$$ What is $a_{15}$? [b]p7.[/b] Consider $5$ points in the plane, no three of which are collinear. Let $n$ be the number of circles that can be drawn through at least three of the points. What are the possible values of $n$? [b]p8.[/b] Find the number of positive integers $n$ satisfying $\lfloor n /2014 \rfloor =\lfloor n/2016 \rfloor$. [b]p9.[/b] Let $f$ be a function taking real numbers to real numbers such that for all reals $x \ne 0, 1$, we have $$f(x) + f \left( \frac{1}{1 - x}\right)= (2x - 1)^2 + f\left( 1 -\frac{1}{ x}\right)$$ Compute $f(3)$. [b]p10.[/b] Alice and Bob split $5$ beans into piles. They take turns removing a positive number of beans from a pile of their choice. The player to take the last bean loses. Alice plays first. How many ways are there to split the piles such that Alice has a winning strategy? [b]p11.[/b] Triangle $ABC$ is an equilateral triangle of side length $1$. Let point $M$ be the midpoint of side $AC$. Another equilateral triangle $DEF$, also of side length $1$, is drawn such that the circumcenter of $DEF$ is $M$, point $D$ rests on side $AB$. The length of $AD$ is of the form $\frac{a+\sqrt{b}}{c}$ , where $b$ is square free. What is $a + b + c$? [b]p12.[/b] Consider the function $f(x) = \max \{-11x- 37, x - 1, 9x + 3\}$ defined for all real $x$. Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with x values $t_1$, $t_2$ and $t_3$ Compute the maximum value of $t_1 + t_2 + t_3$ over all possible $p$. [b]p13.[/b] Circle $J_1$ of radius $77$ is centered at point $X$ and circle $J_2$ of radius $39$ is centered at point $Y$. Point $A$ lies on $J1$ and on line $XY$ , such that A and Y are on opposite sides of $X$. $\Omega$ is the unique circle simultaneously tangent to the tangent segments from point $A$ to $J_2$ and internally tangent to $J_1$. If $XY = 157$, what is the radius of $\Omega$ ? [b]p14.[/b] Find the smallest positive integer $n$ so that for any integers $a_1, a_2,..., a_{527}$,the number $$\left( \prod^{527}_{j=1} a_j\right) \cdot\left( \sum^{527}_{j=1} a^n_j\right)$$ is divisible by $527$. [b]p15.[/b] A circle $\Omega$ of unit radius is inscribed in the quadrilateral $ABCD$. Let circle $\omega_A$ be the unique circle of radius $r_A$ externally tangent to $\Omega$, and also tangent to segments $AB$ and $DA$. Similarly define circles $\omega_B$, $\omega_C$, and $\omega_D$ and radii $r_B$, $r_C$, and $r_D$. Compute the smallest positive real $\lambda$ so that $r_C < \lambda$ over all such configurations with $r_A > r_B > r_C > r_D$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and will paint each of the six sections a solid color. Find the number of ways you can choose to paint each of the six sections if no two adjacent section can be painted with the same color. [asy] size(3cm); draw(unitcircle); draw(scale(0.6)*unitcircle); for(int i = 0; i < 6; ++i){ draw(dir(60*i)--0.6*dir(60*i)); } [/asy]

In the expansion of \[ \left(1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{27}\right)\left(1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{14}\right)^2, \]what is the coefficient of $ x^{28}$? $ \textbf{(A)}\ 195 \qquad \textbf{(B)}\ 196 \qquad \textbf{(C)}\ 224 \qquad \textbf{(D)}\ 378 \qquad \textbf{(E)}\ 405$

How many positive integer tuples $ (x_1,x_2,\dots, x_{13})$ are there satisfying the inequality $x_1+x_2+\dots + x_{13}\leq 2006$? $ \textbf{(A)}\ \frac{2006!}{13!1993!} \qquad\textbf{(B)}\ \frac{2006!}{14!1992!} \qquad\textbf{(C)}\ \frac{1993!}{12!1981!} \qquad\textbf{(D)}\ \frac{1993!}{13!1980!} \qquad\textbf{(E)}\ \text{None of above} $

Wendy randomly chooses a positive integer less than or equal to $2020$. The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ board in which all three colors are used at least once. For example, a $1 \times 1$ red tile followed by a $2 \times 1$ green tile, a $1 \times 1$ green tile, a $2 \times 1$ blue tile, and a $1 \times 1$ green tile is a valid tiling. Note that if the $2 \times 1$ blue tile is replaced by two $1 \times 1$ blue tiles, this results in a different tiling. Find the remainder when $N$ is divided by $1000$.

$N$ teams take part in a league. Every team plays every other team exactly once during the league, and receives 2 points for each win, 1 point for each draw, and 0 points for each loss. At the end of the league, the sequence of total points in descending order $\mathcal{A} = (a_1 \ge a_2 \ge \cdots \ge a_N )$ is known, as well as which team obtained which score. Find the number of sequences $\mathcal{A}$ such that the outcome of all matches is uniquely determined by this information. [I]Proposed by Dominic Yeo, United Kingdom.[/i]

A [i]domino[/i] is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^2$ dominoes can be placed without overlapping on a $2n \times 2n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column.

Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 630 $

Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform [i]stone moves[/i], defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i<j$ and $k<l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively, or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively. Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves. How many different non-equivalent ways can Steve pile the stones on the grid?

Adam has a box with $15$ pool balls in it, numbered from $1$ to $15$, and picks out $5$ of them. He then sorts them in increasing order, takes the four differences between each pair of adjacent balls, and finds exactly two of these differences are equal to $1$. How many selections of $5$ balls could he have drawn from the box? [i]Proposed by Adam Bertelli[/i]

A PUMaC grader is grading the submissions of forty students $s_1, s_2, ..., s_{40}$ for the individual finals round, which has three problems. After grading a problem of student $s_i$, the grader either: $\bullet$ grades another problem of the same student, or $\bullet$ grades the same problem of the student $s_{i-1}$ or $s_{i+1}$ (if $i > 1$ and $i < 40$, respectively). He grades each problem exactly once, starting with the first problem of $s_1$ and ending with the third problem of $s_{40}$. Let $N$ be the number of different orders the grader may grade the students’ problems in this way. Find the remainder when $N$ is divided by $100$.