Let $ABCD$ be a parallelogram. Suppose a line passing through $C$ and lying outside the parallelogram meets $AB$ and $AD$ produced at $E$ and $F$ respectively. Show that \[ AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF . \]

i) If $b=\dfrac{a^2+ \dfrac{1}{a^2}}{a^2-\dfrac{1}{a^2}}$ , express $c=\dfrac{a^4+\dfrac{1}{a^4}}{a^4-\dfrac{1}{a^4}}$ , in terms of $b$. ii) If $k= \frac{x^{n}+\dfrac{1}{x^{n}}}{x^{n}-\dfrac{1}{x^{n}}}$, express $m= \frac{x^{2n}+\dfrac{1}{x^{2n}}}{x^{2n}-\dfrac{1}{x^{2n}}}$ in terms of $k$.

Find the least positive integer which is a multiple of $13$ and all its digits are the same. [i](Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)[/i]

Anton is playing a game with shapes. He starts with a circle $\omega_1$ of radius $1$, and to get a new circle $\omega_2$, he circumscribes a square about $\omega_1$ and then circumscribes circle $\omega_2$ about that square. To get another new circle $\omega_3$, he circumscribes a regular octagon about circle $\omega_2$ and then circumscribes circle $\omega_3$ about that octagon. He continues like this, circumscribing a $2n$-gon about $\omega_{n-1}$ and then circumscribing a new circle $\omega_n$ about the $2n$-gon. As $n$ increases, the area of $\omega_n$ approaches a constant $A$. Compute $A$.

Let $n\in\mathbb{Z}$, $n\geq 3$. A matrix $A\in\mathcal{M}_n(\mathbb{C})$ is said to have property $(\mathcal{P})$ if $\det(A+X_{ij})=\det(A+X_{ji})$, for all $i,j\in\{1,2,\dots ,n\}$, where $X_{ij}\in\mathcal{M}_n(\mathbb{C})$ is the matrix with $1$ on position $(i,j)$ and $0$ otherwise. [list=a] [*] Show that if $A\in\mathcal{M}_n(\mathbb{C})$ has property $(\mathcal{P})$ and $\det(A)\neq 0$, then $A=A^T$. [*] Give an example of a matrix $A\in\mathcal{M}_n(\mathbb{C})$ with property $(\mathcal{P})$ such that $A\neq A^T$.

The sequences $(x_n), (y_n), (z_n)$ are given by $x_{n+1}=y_n +\frac{1}{x_n}$,$ y_{n+1}=z_n +\frac{1}{y_n}$,$z_{n+1}=x_n +\frac{1}{z_n} $ for $n \ge 0$ where $x_0,y_0, z_0$ are given positive numbers. Prove that these sequences are unbounded.

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

In a sequence of numbers, a term is called [i]golden [/i] if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1, 2, 3, . . . , 2021$?

Find the number of functions $f(x)$ from $\{1,2,3,4,5\}$ to $\{1,2,3,4,5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1,2,3,4,5\}$.

Let $x, y$ be positive integers such that $$ x^4=(x-1)\left(y^3-23\right)-1 . $$ Find the maximum possible value of $x+y$.

Michael the Mouse stands in a circle with $11$ other mice. Eshaan the Elephant walks around the circle, squashing every other non-squashed mouse he comes across. If it takes Eshaan $1$ minute ($60$ seconds) to complete one circle and he walks at a constant rate, what is the maximum length of time in seconds from when the first mouse is squashed that Michael can survive? [i]2015 CCA Math Bonanza Lightning Round #3.3[/i]

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$ $(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$

Prove that $ |f(x)|\le |f(0)| +\int_0^x |f(t) +f'(t)|dt , $ for any nonnegative real numbers $ x, $ and functions $f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ of class $ \mathcal{C}^1. $

Find the value of $c$ such that the system of equations \begin{align*}|x+y|&=2007,\\|x-y|&=c\end{align*} has exactly two solutions $(x,y)$ in real numbers. $\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l} \textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\ \textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\ \textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\ \textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\ \textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\ \textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\ \textbf{(S) }18&\textbf{(T) }223&\textbf{(U) }678\\\\ \textbf{(V) }2007 & &\end{array}$

A magician and his assistent are performing the following trick.There is a row of 12 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistent knows which boxes contain coins. The magician returns, and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects 4 boxes, which are simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always succesfully perform the trick.

Let $AB$ be a segment of length $1$. Several particles start moving simultaneously at constant speeds from $A$ up to$ B$. As soon as a particle reaches $B$ , turns around and goes to $A$; when it reaches $A$, begins to move again towards $ B$ , and so on indefinitely. Find all rational numbers$ r>1$ such that there exists an instant $t$ with the following property: For each $n\ge 1$ , if $n+1$ particles with constant speeds $1,r,r^2,…,r^n$ move as described, at instant $t$, they all lie at the same interior point of segment $AB$.

On the board we write a series of $n$ numbers, where $n \geq 40$, and each one of them is equal to either $1$ or $-1$, such that the following conditions both hold: (i) The sum of every $40$ consecutive numbers is equal to $0$. (ii) The sum of every $42$ consecutive numbers is not equal to $0$. We denote by $S_n$ the sum of the $n$ numbers of the board. Find the maximum possible value of $S_n$ for all possible values of $n$.

Let $N$ be the number of ordered lists of $9$ positive integers $(a,b,c,d,e,f,g,h,i)$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}+\frac{1}{g}+\frac{1}{h}+\frac{1}{i}=1.$$ Determine whether $N$ is even or odd.

Let $x_1$ and $x_2$ be roots of the equation $x^2 - 6x + 1 = 0$. Prove that for any integer $n \ge 1$ the number $x_1^n + x_2^n$ is integer and is not divisible by $5$.

Let $n$ be a natural number. We have $n$ colors. Each of the numbers $1, 2, 3,... , 1000$ was colored with one of the $n$ colors. It is known that, for any two distinct numbers, if one divides the other then these two numbers have different colors. Determine the smallest possible value of $n$.

In triangle $\triangle ABC$, $BC>AC$, $I_B$ is the $B$-excenter, the line through $C$ parallel to $AB$ meets $BI_B$ at $F$. $M$ is the midpoint of $AI_B$ and the $A$-excircle touches side $AB$ at $D$. Point $E$ satisfies $\angle BAC=\angle BDE, DE=BC$, and lies on the same side as $C$ of $AB$. Let $EC$ intersect $AB,FM$ at $P,Q$ respectively. Prove that $P,A,M,Q$ are concyclic.

Given a regular nonagon $A_1A_2A_3A_4A_5A_6A_7A_8A_9$ with side length $1$. Diagonals $A_3A_7$ and $A_4A_8$ intersect at point $P$. Find the length of segment $P A_1$.

Let $T$ be the triangle in the coordinate plane with vertices $\left(0,0\right)$, $\left(4,0\right)$, and $\left(0,3\right)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.) $\textbf{(A) } 12\qquad\textbf{(B) } 15\qquad\textbf{(C) }17 \qquad\textbf{(D) }20 \qquad\textbf{(E) }25$

[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square. [b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$. [b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img] [b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Anne multiplies each two-digit number by $588$ in turn, and writes down the so-obtained products. How many perfect squares does she write down?