Consider a table with $n$ rows and $2n$ columns. we put some blocks in some of the cells. After putting blocks in the table we put a robot on a cell and it starts moving in one of the directions right, left, down or up. It can change the direction only when it reaches a block or border. Find the smallest number $m$ such that we can put $m$ blocks on the table and choose a starting point for the robot so it can visit all of the unblocked cells. (the robot can't enter the blocked cells.) Proposed by Seyed Mohammad Seyedjavadi and Alireza Tavakoli

Let $a,b,c$ be positive reals and sequences $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ defined by $a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}$ for all $k=0,1,2,...$. Prove that $\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty$.

The [i]base factorial[/i] number system is a unique representation for positive integers where the $n$th digit from the right ranges from $0$ to $n$ inclusive and has place value $n!$ for all $n \ge 1.$ For instance, $71$ can be written in base factorial as $2321_{!} = 2 \cdot 4! + 3 \cdot 3! + 2 \cdot 2! + 1 \cdot 1!.$ Let $S_{!}(n)$ be the base $10$ sum of the digits of $n$ when $n$ is written in base factorial. Compute $\sum_{n=1}^{700} S_{!}(n)$ (expressed in base $10$).

Sita and Geeta are two sisters. If Sita's age is written after Geeta's age a four digit perfect square (number) is obtained. If the same exercise is repeated after 13 years another four digit perfect square (number) will be obtained. What is the sum of the present ages of Sita and Geeta?

There are $ n\ge 1 $ ordered bulbs controlled by $ n $ ordered switches such that the $ k\text{-th} $ switch controls the $ k\text{-th} $ bulb and also the $ j\text{-th} $ bulb if and only if the $ j\text{-th} $ switch controls the $ k\text{-th} $ bulb, for any $ 1\le k,j\le n. $ If all bulbs are off, show that it can be chosen some switches such that, if pushed simmultaneously, the bulbs turn all on.

The $53$-digit number $$37,984,318,966,591,152,105,649,545,470,741,788,308,402,068,827,142,719$$ can be expressed as $n^21$ where $n$ is a positive integer. Find $n$.

Let circles $s_1$ and $s_2$ meet at points $A$ and $B$. Consider all lines passing through $A$ and meeting the circles for the second time at points $P_1$ and $P_2$ respectively. Construct by a compass and a ruler a line such that $AP_1.AP_2$ is maximal.

Let an ordered pair of positive integers $(m, n)$ be called [i]regimented[/i] if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $\left(2016^{2016}, N\right)$ is regimented. Compute the largest positive integer $v$ such that $2^v$ divides the difference $2016^{2016}-N$. [i]Proposed by Ashwin Sah[/i]

The solution set to the inequality $\sqrt{\log_2 x-1}+\frac{1}{2}\log_{\frac{1}{2}}x^3+2>0$ is $\text{(A)}[2,3)\qquad\text{(B)}(2,3]\qquad\text{(C)}[2,4)\qquad\text{(D)}(2,4]$

a) * In a rectangle of area $5$ sq. units, $9$ rectangles of area $1$ are arranged. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$ b) In a rectangle of area $5$ sq. units, lie $9$ arbitrary polygons each of area $1$. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$

Prove that if $ n $ is a natural number greater than $ 1 $, then $$ \cos \frac{2\pi}{n} + \cos \frac{4\pi}{n} + \cos \frac{6\pi}{n} + \ldots + \cos \frac{2n \pi}{n} = 0.$$

Show that when a prime number is divided by $30$, the remainder is $1$ or a prime number. Shows that if it is divided by $60$ or $90$ the same thing does not happen.

Consider two parallel lines $a$ and $b$. The circles $C,C_1$ are tangent to each other and to the line $a$. The circles $C,C_2$ are tangent to each other and to the line $b$. The circles $C_1,C_2$ are tangent to each other, have radii $R_1=9,R_2=16$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy82L2E1Y2NjMTY0NTNjMzExYmRmOTZjYmZlMWMwNzE4YmNlM2I0YTNkLnBuZw==&rn=Sm9zZXBoIDIwMjAucG5n[/img] What is the radius $R$ of the circle $C$? [i]Proposed by Ovidiu Bagdasar[/i]

A dealer bought $n$ radios for $d$ dollars, $d$ a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $\$8$ on each radio sold. If the overall profit was $\$72$, then the least possible value of $n$ for the given information is: $\textbf{(A)}\ 18\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 11$

All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Marcy could have? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce? [asy] //diagram by pog size(8.5cm); usepackage("mathptmx"); defaultpen(mediumgray*0.5+gray*0.5+linewidth(0.63)); add(grid(6,6)); label(scale(0.7)*"$1$", (1,-0.3), black); label(scale(0.7)*"$2$", (2,-0.3), black); label(scale(0.7)*"$3$", (3,-0.3), black); label(scale(0.7)*"$4$", (4,-0.3), black); label(scale(0.7)*"$5$", (5,-0.3), black); label(scale(0.7)*"$1$", (-0.3,1), black); label(scale(0.7)*"$2$", (-0.3,2), black); label(scale(0.7)*"$3$", (-0.3,3), black); label(scale(0.7)*"$4$", (-0.3,4), black); label(scale(0.7)*"$5$", (-0.3,5), black); label(scale(0.75)*rotate(90)*"Price (dollars)", (-1,3.2), black); label(scale(0.75)*"Weight (ounces)", (3.2,-1), black); dot((1,1.2),black); dot((1,1.7),black); dot((1,2),black); dot((1,2.8),black); dot((1.5,2.1),black); dot((1.5,3),black); dot((1.5,3.3),black); dot((1.5,3.75),black); dot((2,2),black); dot((2,2.9),black); dot((2,3),black); dot((2,4),black); dot((2,4.35),black); dot((2,4.8),black); dot((2.5,2.7),black); dot((2.5,3.7),black); dot((2.5,4.2),black); dot((2.5,4.4),black); dot((3,2.5),black); dot((3,3.4),black); dot((3,4.2),black); dot((3.5,3.8),black); dot((3.5,4.5),black); dot((3.5,4.8),black); dot((4,3.9),black); dot((4,5.1),black); dot((4.5,4.75),black); dot((4.5,5),black); dot((5,4.5),black); dot((5,5),black); [/asy] $\textbf{(A)} ~1\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~3\qquad\textbf{(D)} ~4\qquad\textbf{(E)} ~5\qquad$

Let $ABCD$ be a quadrilateral satisfying $\angle BCD=\angle CDA$. Suppose rays $AD$ and $BC$ meet at $E$, and let $\Gamma$ be the circumcircle of $ABE$. Let $\Gamma_1$ be a circle tangent to ray $CD$ past $D$ at $W$, segment $AD$ at $X$, and internally tangent to $\Gamma$. Similarly, let $\Gamma_2$ be a circle tangent to ray $DC$ past $C$ at $Y$, segment $BC$ at $Z$, and internally tangent to $\Gamma$. Let $P$ be the intersection of $WX$ and $YZ$, and suppose $P$ lies on $\Gamma$. If $F$ is the $E$-excenter of triangle $ABE$, and $AB=544$, $AE=2197$, $BE=2299$, then find $m+n$, where $FP=\tfrac{m}{n}$ with $m,n$ relatively prime positive integers. [i]Proposed by Michael Kural[/i]

Evaluate \[\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx\]

All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is: (a) an integer; (b) the square of an integer. (A Kanel)

Alice and Bob play the following game. They write some fractions of the form $1/n$, where $n{}$ is positive integer, onto the blackboard. The first move is made by Alice. Alice writes only one fraction in each her turn and Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn and so on. Bob wants to make the sum of all the fractions on the board to be an integer number after some turn. Can Alice prevent this? [i]Andrey Arzhantsev[/i]

Prove that there exists a positive integer $N$ such that for every polynomial $P(x)$ of degree $2019$, there exist $N$ linear polynomials $p_1,p_2, \ldots p_N$ such that $P(x)=p_1(x)^{2019}+p_2(x)^{2019}+ \ldots + p_N(x)^{2019}$. (Assume all polynomials in this problem have real coefficients, and leading coefficients cannot be zero.)

A convex hexagon is circumscribed about a circle of radius $1$. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r.$

In kindergarten, nurse took $n>1$ identical cardboard rectangles and distributed them to $n$ children; every child got one rectangle. Every child cut his (her) rectangle into several identical squares (squares of different children could be different). Finally, the total number of squares was prime. Prove that initial rectangles was squares.

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.