From $1,2,\cdots,14$, take out three numbers $a_1<a_2<a_3$, satisfying that $a_2-a_1\geq3,a_3-a_2\geq3$. Then the number of different ways of taking out numbers is________.

If $a_1, a_2, \ldots$ is a sequence of real numbers such that for all $n$, $$\sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1,$$ find the smallest $n$ such that $a_n < \frac{1}{2018}$.

Let $O$ be the center of the circumcircle, and $AD$ be the angle bisector of the acute triangle $ABC$. The perpendicular drawn from point $D$ on the line $AO$ ​​intersects the line $AC$ at the point $P$. Prove that $AP = AB$.

Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$. Proposed by Audrey Chun

Let $ \text{ABCD} $ be a cyclic quadrilateral, with $ \text{AB} = 7 $, $ \text{BC} = 11 $, $ \text{CD} = 13 $, and $ \text{DA} = 17 $. Let the incircle of $ \text{ABD} $ hit $ \text{BD} $ at $ \text{R} $ and the incircle of $ \text{CBD} $ hit $ \text{BD} $ at $ \text{S} $. What is $ \text{RS} $?

How many complex numbers satisfy the equation $z^{5}=\overline{z}$, where $\overline{z}$ is the conjugate of the complex number $z$? $\textbf{(A)}~2\qquad\textbf{(B)}~3\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$

Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)} {(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$ holds for all triples of distinct integers $x, y, z$. When does equality hold?

Let $a_1, a_2, \ldots , a_n, \ldots $ be a sequence of real numbers such that $0 \leq a_n \leq 1$ and $a_n - 2a_{n+1} + a_{n+2} \geq 0$ for $n = 1, 2, 3, \ldots$. Prove that \[0 \leq (n + 1)(a_n - a_{n+1}) \leq 2 \qquad \text{ for } n = 1, 2, 3, \ldots\]

Let $a_1, a_2, \ldots, a_n$ be real numbers such that $\sum_{i=1}^n a_i = n$, $\sum_{i = 1}^n a_i^2 = 2n$, $\sum_{i=1}^n a_i^3 = 3n$. (i) Find the largest constant $C$, such that for all $n \geqslant 4$, \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C. \] (ii) Prove that there exists a positive constant $C_2$, such that \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C + C_2 n^{-\frac 32}, \]where $C$ is the constant determined in (i).

Incircle $\omega$ of triangle $ABC$ is tangent to sides $CB$ and $CA$ at $D$ and $E$, respectively. Point $X$ is the reflection of $D$ with respect to $B$. Suppose that the line $DE$ is tangent to the $A$-excircle at $Z$. Let the circumcircle of triangle $XZE$ intersect $\omega$ for the second time at $K$. Prove that the intersection of $BK$ and $AZ$ lies on $\omega$. Proposed by Mahdi Etesamifard and Alireza Dadgarnia

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

Circle $\omega_1$ of radius $1$ and circle $\omega_2$ of radius $2$ are concentric. Godzilla inscribes square $CASH$ in $\omega_1$ and regular pentagon $MONEY$ in $\omega_2$. It then writes down all 20 (not necessarily distinct) distances between a vertex of $CASH$ and a vertex of $MONEY$ and multiplies them all together. What is the maximum possible value of his result?

Rectangle $ABCD$ has side lengths $AB = 3$ and $BC = 7$. Let $E$ be a point on $BC$, and let $F$ be the intersection of $DE$ and $AC$. Given that $[CDF] = 4$, find $\frac{DF}{FE}$ .

Below is a $4\times4$ grid. We wish to fill in the grid such that each row, each column, and each $2\times2$ square outlined by the double lines contains the digits 1 through 4. The first row has already been filled in. Find, with proof, the number of ways we can complete the rest of the grid. \[ \begin{tabular}{||c|c||c|c||}\hline\hline 1 & 2 & 3 & 4\\ \hline &&&\\ \hline\hline &&&\\ \hline &&&\\ \hline\hline \end{tabular} \]

There are $n$ distinct points on a circumference. Choose one of the points. Connect this point and the $m$th point from the chosen point counterclockwise with a segment. Connect this $m$th point and the $m$th point from this $m$th point counterclockwise with a segment. Repeat such steps until no new segment is constructed. From the intersections of the segments, let the number of the intersections - which are in the circle - be $I$. Answer the following questions ($m$ and $n$ are positive integers that are relatively prime and they satisfy $6 \leq 2m < n$). 1) When the $n$ points take different positions, express the maximum value of $I$ in terms of $m$ and $n$. 2) Prove that $I \geq n$. Prove that there is a case, which is $I=n$, when $m=3$ and $n$ is arbitrary even number that satisfies the condition.

Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad \textbf{(E) }18$