For $n \in \mathbb{N}$, let $a_n = \int _0 ^{1/\sqrt{n}} | 1 + e^{it} + e^{2it} + \dots + e^{nit} | \ dt$. Determine whether the sequence $(a_n) = a_1, a_2, \dots$ is bounded.

Let $A = \frac{1 \cdot 3 \cdot 5\cdot ... \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot (2n)}$ Prove that in the infinite sequence $A, 2A, 4A, 8A, ..., 2^k A, ….$ only integers will be observed, eventually.

Source: 2018 Canadian Open Math Challenge Part A Problem 2 ----- Let $v$, $w$, $x$, $y$, and $z$ be five distinct integers such that $45 = v\times w\times x\times y\times z.$ What is the sum of the integers?

Two cubes with edge length $3$ and two cubes with edge length $4$ sit on plane $P$ so that the four cubes share a vertex, and the two larger cubes share no faces with each other as shown below. The cube vertices that do not touch $P$ or any of the other cubes are labeled $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $H$. The four cubes lie inside a right rectangular pyramid whose base is on $P$ and whose triangular sides touch the labeled vertices with one side containing vertices $A$, $B$, and $C$, another side containing vertices $D$, $E$, and $F$, and the two other sides each contain one of $G$ and $H$. Find the volume of the pyramid.

[asy] import cse5; size(180); real a=4, b=3; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP("A",A)--MP("B",B,N)--MP("C",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP("R",IP(CP((X+C)/2,C),B--C),NW)--MP("Q",IP(CP((X+C)/2,C),A--C+(0.1,0)))); //Credit to chezbgone2 for the diagram[/asy] In $\triangle ABC$, $AB = 10~ AC = 8$ and $BC = 6$. Circle $P$ is the circle with smallest radius which passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$ , of circle $P$ with sides $AC$ and $BC$, respectively. The length of segment $QR$ is $\textbf{(A) }4.75\qquad\textbf{(B) }4.8\qquad\textbf{(C) }5\qquad\textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }3\sqrt{3}$

Segments $AD=10$, $BE=6$, $CF=24$ are drawn from the vertices of triangle $ABC$, each perpendicular to a straight line $RS$, not intersecting the triangle. Points $D$, $E$, $F$ are the intersection points of $RS$ with the perpendiculars. If $x$ is the length of the perpendicular segment $GH$ drawn to $RS$ from the intersection point $G$ of the medians of the triangle, then $x$ is: $\textbf{(A)}\ \frac{40}{3}\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ \frac{56}{3}\qquad \textbf{(D)}\ \frac{80}{3}\qquad \textbf{(E)}\ \text{undetermined}$

In the beginning there are $100$ integers in a row on the blackboard. Kain and Abel then play the following game: A [i]move[/i] consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers $1,2,\dots,100$ and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add $1$ to each of the chosen numbers or instead subtract $1$ from of the numbers. After that the next move begins, and so on. If there are at least $98$ numbers on the blackboard that are divisible by $4$ after a move, then Kain has won. Prove that Kain can force a win in a finite number of moves.

Roger the ant is traveling on a coordinate plane, starting at $(0,0)$. Every second, he moves from one lattice point to a different lattice point at distance $1$, chosen with equal probability. He will continue to move until he reaches some point $P$ for which he could have reached more quickly had he taken a different route. For example, if he goes from $(0,0)$ to $(1,0)$ to $(1,1)$ to $(1,2)$ to $(0,2)$, he stops at because he could have gone from $(0,0)$ to $(0,1)$ to $(0,2)$ in only $2$ seconds. The expected number of steps Roger takes before he stops can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$.

Let $ABC$ be an acute triangle such that $\overline{AB}=\overline{AC}$. Let $M, L, N$ be the midpoints of segment $BC, AM, AC$, respectively. The circumcircle of triangle $AMC$, denoted by $\Omega$, meets segment $AB$ at $P(\neq A)$, and the segment $BL$ at $Q$. Let $O$ be the circumcenter of triangle $BQC$. Suppose that the lines $AC$ and $PQ$ meet at $X$, $OB$ and $LN$ meet at $Y$, and $BQ$ and $CO$ meets at $Z$. Prove that the points $X, Y, Z$ are collinear.

Alice and Bob are playing the following game: They take turns writing on the board natural numbers not exceeding $2018$ (to write the number twice is forbidden). Alice begins. A player wins if after his or her move there appear three numbers on the board which are in arithmetic progression. Which player has a winning strategy?

Let (i) $a_1,a_2,a_3$ is an arithmetic progression and $a_1+a_2+a_3=18$ (ii) $b_1,b_2,b_3$ is a geometric progression and $b_1b_2b_3=64$ If $a_1+b_1,a_2+b_2,a_3+b_3$ are all positive integers and it is a ageometric progression, then find the maximum value of $a_3$.

Cast a dice $n$ times. Denote by $X_1,\ X_2,\ \cdots ,\ X_n$ the numbers shown on each dice. Define $Y_1,\ Y_2,\ \cdots,\ Y_n$ by \[Y_1=X_1,\ Y_k=X_k+\frac{1}{Y_{k-1}}\ (k=2,\ \cdots,\ n)\] Find the probability $p_n$ such that $\frac{1+\sqrt{3}}{2}\leq Y_n\leq 1+\sqrt{3}.$ 35 points

Prove that for all $n\in \mathbb{N}$ there exist natural numbers $a_1,a_2,...,a_n$ such that: $(i)a_1>a_2>...>a_n$ $(ii)a_i|a^2_{i+1},\forall i\in\{1,2,...,n-1\}$ $(iii)a_i\nmid a_j,\forall i,j\in \{1,2,...,n\},i\neq j$

Does there exist an operation $*$ on the set of all integers such that the following conditions hold simultaneously: $(1)$ for all integers $x,y,z$, $(x*y)*z=x*(y*z)$; $(2)$ for all integers $x$ and $y$, $x*x*y=y*x*x=y$?

In a trapezoid $ABCD$ with $AB<CD$ and $AB \parallel CD$, the diagonals intersect each other at $E$. Let $F$ be the midpoint of the arc $BC$ (not containing the point $E$) of the circumcircle of the triangle $EBC$. The lines $EF$ and $BC$ intersect at $G$. The circumcircle of the triangle $BFD$ intersects the ray $[DA$ at $H$ such that $A \in [HD]$. The circumcircle of the triangle $AHB$ intersects the lines $AC$ and $BD$ at $M$ and $N$, respectively. $BM$ intersects $GH$ at $P$, $GN$ intersects $AC$ at $Q$. Prove that the points $P, Q, D$ are collinear.

Q. A light source at the point $(0, 16)$ in the co-ordinate plane casts light in all directions. A disc(circle along ith it's interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m, n$ are positive integers and $n$ is squarefree. Find $m + n$.

Let $a,b,c,d$ be four non-negative numbers satisfying \[ a+b+c+d=1. \] Prove the inequality \[ a \cdot b \cdot c + b \cdot c \cdot d + c \cdot d \cdot a + d \cdot a \cdot b \leq \frac{1}{27} + \frac{176}{27} \cdot a \cdot b \cdot c \cdot d. \]

Square $ABCD$ has side length $4$. Points $P$ and $Q$ are located on sides $BC$ and $CD$, respectively, such that $BP = DQ = 1$. Let $AQ$ intersect $DP$ at point $X$. Compute the area of triangle $P QX$.

Let $ n $ be a positive integer and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $ a_1^2+a_2^2+\cdots +a_n^2=1. $ Show that $$ \sum_{1\le ij\le n} a_ia_j<2\sqrt n. $$ [i]Russian math competition[/i]

Subsets $S$ of the first 3$5$ positive integers $\{1, 2, 3, ..., 35\}$ are called [i]contrived [/i] if $S$ has size $4$ and the sum of the squares of the elements of $S$ is divisible by $7$. Find the number of contrived sets.

Let $ x, y, z $ be positive real numbers. Prove that \[ \frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1 \]

The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is: ${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3}\qquad\textbf{(E)}\ \text{More than three, but finite} } $

$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. (See diagram). Given $EC=1$, find the radius of the circle. [asy] size(6cm); pair O = (0,0), B = dir(110), D = dir(30), E = 0.4 * B + 0.6 * D, C = intersectionpoint(O--2*E, unitcircle); draw(unitcircle); draw(O--C); draw(B--D); dot(O); dot(B); dot(C); dot(D); dot(E); label("$B$", B, B); label("$C$", C, C); label("$D$", D, D); label("$E$", E, dir(280)); label("$O$", O, dir(270)); [/asy]

[color=darkred]A row of a matrix belonging to $\mathcal{M}_n(\mathbb{C})$ is said to be [i]permutable[/i] if no matter how we would permute the entries of that row, the value of the determinant doesn't change. Prove that if a matrix has two [i]permutable[/i] rows, then its determinant is equal to $0$ .[/color]

For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even? [i]Proposed by Andrew Wen[/i]