The diagonal of square I is $a+b$. The perimeter of square II with [i]twice[/i] the area of I is: $ \textbf{(A)}\ (a+b)^2\qquad\textbf{(B)}\ \sqrt{2}(a+b)^2\qquad\textbf{(C)}\ 2(a+b)\qquad\textbf{(D)}\ \sqrt{8}(a+b) \qquad$ $\textbf{(E)}\ 4(a+b) $

Let $d(n)$ be the number of positive divisors of a positive integer $n$. Let $\mathbb{N}$ be the set of all positive integers. Say that a bijection $F$ from $\mathbb{N}$ to $\mathbb{N}$ is [i]divisor-friendly[/i] if $d(F(mn)) = d(F(m)) d(F(n))$ for all positive integers $m$ and $n$. (Note: A bijection is a one-to-one, onto function.) Does there exist a divisor-friendly bijection? Prove or disprove.

During a day $2016$ customers visited the store. Every customer has been only once at the store(a customer enters the store,spends some time, and leaves the store). Find the greatest integer $k$ that makes the following statement always true. We can find $k$ customers such that either all of them have been at the store at the same time, or any two of them have not been at the same store at the same time.

Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side $AB$ and is tangent to the sides $AC$ and $BC$. The circles $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.

In a tennis tournament, any two of the $n$ participants played a match (the winner of a match gets $1$ point, the loser gets $0$). The scores after the tournament were $r_1\le r_2\le\ldots\le r_n$. A match between two players is called wrong if after it the winner has a score less than or equal to that of the loser. Consider the set $I=\{i|r_1\ge i\}$. Prove that the number of wrong matches is not less than $\sum_{i\in I}(r_i-i+1)$, and show that this value is realizable

[b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.

The positive integers $a $ and $b $ satisfy the sistem $\begin {cases} a_{10} +b_{10} = a \\a_{11}+b_{11 }=b \end {cases} $ where $ a_1 <a_2 <\dots $ and $ b_1 <b_2 <\dots $ are the positive divisors of $a $ and $b$ . Find $a$ and $b $ .

Is there a positive integer $ k$ such that none of the digits $ 3,4,5,6$ appear in the decimal representation of the number $ 2002!\cdot k$?

Show that for all real numbers $x,y,z$ such that $x + y + z = 0$ and $xy + yz + zx = -3$, the expression $x^3y + y^3z + z^3x$ is a constant.

Find all possibilities: how many acute angles can there be in a convex polygon?

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

Given a positive integer $n$. The number $2n$ has $28$ positive factors, while the number $3n$ has $30$ positive factors. Find the number of positive divisors of $6n$.

[b]p1[/b]. Points $P_1,P_2,P_3,... ,P_n$ lie on a plane such that $P_aP_b = 1$,$P_cP_d = 2$, and $P_eP_f = 2018$ for not necessarily distinct indices $a,b,c,d,e, f \in \{1, 2,... ,n\}$. Find the minimum possible value of $n$. [b]p2.[/b] Find the coefficient of the $x^2y^4$ term in the expansion of $(3x +2y)^6$. [b]p3.[/b] Find the number of positive integers $n < 1000$ such that $n$ is a multiple of $27$ and the digit sum of $n$ is a multiple of $11$. [b]p4.[/b] How many times do the minute hand and hour hand of a $ 12$-hour analog clock overlap in a $366$-day leap year? [b]p5.[/b] Find the number of ordered triples of integers $(a,b,c)$ such that $(a +b)(b +c)(c + a) = 2018$. [b]p6.[/b] Let $S$ denote the set of the first $2018$ positive integers. Call the score of a subset the sum of its maximal element and its minimal element. Find the sum of score $(x)$ over all subsets $s \in S$ [b]p7.[/b] How many ordered pairs of integers $(a,b)$ exist such that $1 \le a,b \le 20$ and $a^a$ divides $b^b$? [b]p8.[/b] Let $f$ be a function such that for every non-negative integer $p$, $f (p)$ equals the number of ordered pairs of positive integers $(a,n)$ such that $a^n = a^p \cdot n$. Find $\sum^{2018}_{p=0}f (p)$. [b]p9.[/b] A point $P$ is randomly chosen inside a regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$. What is the probability that the projections of $P$ onto the lines $\overleftrightarrow{A_i A_{i+1}}$ for $i = 1,2,... ,8$ lie on the segments $\overline{A_iA_{i+1}}$ for $i = 1,2,... ,8$ (where indices are taken $mod \,\, 8$)? [b]p10. [/b]A person keeps flipping an unfair coin until it flips $3$ tails in a row. The probability of it landing on heads is $\frac23$ and the probability it lands on tails is $\frac13$ . What is the expected value of the number of the times the coin flips? PS. You had better use hide for answers.

(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove that $n$ is a power of $2$; (b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such that $2^n-1$ divides $m^2 + 9$.

Let $D_n$ denote the value of the $(n -1) \times (n - 1)$ determinant $$ \begin{pmatrix} 3 & 1 &1 & \ldots & 1\\ 1 & 4 &1 & \ldots & 1\\ 1 & 1 & 5 & \ldots & 1\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 1 & 1 & \ldots & n+1 \end{pmatrix}.$$ Is the set $\left\{ \frac{D_n }{n!} \, | \, n \geq 2\right\}$ bounded?

A uniform pool ball of radius $r$ and mass $m$ begins at rest on a pool table. The ball is given a horizontal impulse $J$ of fixed magnitude at a distance $\beta r$ above its center, where $-1 \le \beta \le 1$. The coefficient of kinetic friction between the ball and the pool table is $\mu$. You may assume the ball and the table are perfectly rigid. Ignore effects due to deformation. (The moment of inertia about the center of mass of a solid sphere of mass $m$ and radius $r$ is $I_{cm} = \frac{2}{5}mr^2$.) [asy] size(250); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((0,0),1),gray(.8)); draw((-3,-1)--(3,-1)); draw((-2.4,0.1)--(-2.4,0.6),EndArrow); draw((-2.5,0)--(2.5,0),dashed); draw((-2.75,0.7)--(-0.8,0.7),EndArrow); label("$J$",(-2.8,0.7),W); label("$\beta r$",(-2.3,0.35),E); draw((0,-1.5)--(0,1.5),dashed); draw((1.7,-0.1)--(1.7,-0.9),BeginArrow,EndArrow); label("$r$",(1.75,-0.5),E); [/asy] (a) Find an expression for the final speed of the ball as a function of $J$, $m$, and $\beta$. (b) For what value of $\beta$ does the ball immediately begin to roll without slipping, regardless of the value of $\mu$?

Alice and Billy are playing a game on a number line. They both start at $0$. Each turn, Alice has a $\frac{1}{2}$ chance of moving $1$ unit in the positive direction, and a $\frac{1}{2}$ chance of moving $1$ unit in the negative direction, while Billy has a $\frac{2}{3}$ chance of moving $1$ unit in the positive direction, and a $\frac{1}{3}$ chance of moving $1$ unit in the negative direction. Alice and Billy alternate turns, with Alice going first. If a player reaches $2$, they win and the game ends, but if they reach $-2$, they lose and the other player wins, and the game ends. What is the probability that Billy wins? [i]2018 CCA Math Bonanza Lightning Round #4.4[/i]

The excircle of a triangle $ABC$ corresponding to $A$ touches the side $BC$ at $M$, and the point on the incircle diametrically opposite to its point of tangency with $BC$ is denoted by $N$. Prove that $A,M,$ and $N$ are collinear.

Given a prime $p$, show that there exist two integers $a, b$ which satisfies the following. For all integers $m$, $m^3+ 2017am+b$ is not a multiple of $p$.

Minor base $BC$ of trapezoid $ABCD$ is equal to side $AB$, and diagonal $AC$ is equal to base $AD$. The line passing through B and parallel to $AC$ intersects line $DC$ in point $M$. Prove that $AM$ is the bisector of angle $\angle BAC$. A.Blinkov, Y.Blinkov

The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to rooms adjacent by side. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess rook (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 1$)?

Let $ ABC$ be an equilateral triangle. Let $ \Omega$ be its incircle (circle inscribed in the triangle) and let $ \omega$ be a circle tangent externally to $ \Omega$ as well as to sides $ AB$ and $ AC$. Determine the ratio of the radius of $ \Omega$ to the radius of $ \omega$.

For the equation \[ (3x^2+y^2-4y-17)^3-(2x^2+2y^2-4y-6)^3=(x^2-y^2-11)^3, \] determine its solutions $(x, y)$ where both $x$ and $y$ are integers. Prove that your answer lists all the integer solutions.

It is given $4$ circles in a plane and every one of them touches the other three as shown: [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC82L2FkYWQ5NThhMWRiMjAwZjYxOWFhYmE1M2YzZDU5YWI2N2IyYzk2LnBuZw==&rn=a3J1Z292aS5wbmc=[/img] Biggest circle has radius $2$, and every one of the medium has $1$. Find out the radius of fourth circle.

The plane is divided in $1\times1$ squares. In each square there is a real number such that it is the arithmetic mean of the four adjacent squares (with a common side). In a square there is $2024.$ Is it possible for $2024^{2024}$ to be written in another square if all the numbers are: a) nonnegative integers; b) integers?