13) A massless rope passes over a frictionless pulley. Particles of mass $M$ and $M + m$ are suspended from the two different ends of the rope. If $m = 0$, the tension $T$ in the pulley rope is $Mg$. If instead the value m increases to infinity, the value of the tension does which of the following? A) stays constant B) decreases, approaching a nonzero constant C) decreases, approaching zero D) increases, approaching a finite constant E) increases to infinity

A triangle has two sides of lengths $1984$ and $2016$. Find the maximum possible area of the triangle. [i]Proposed by Nathan Ramesh

Each of the $10$ dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?

If $\sqrt{x+2}=2$, then $(x+2)^2$ equals $\text{(A)}\ \sqrt{2} \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 16$

The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?

[color=darkred]For any $ m \in \mathbb{N}^*$ solve the ecuation \[ \left\{\left( x \plus{} \frac {1}{m}\right) ^3\right\} \equal{} x^3 \] [/color]

Let $\ell_1$ and $\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\ell_1$ while points $C$ and $D$ lie on $\ell_2$ such that $\angle BAC = 30^\circ$ and $\angle ABD = 60^\circ$. The minimum value of $AD + BC$ is $a\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a + b$.

Let $n \ge 3$ be a natural number. Determine the number $a_n$ of all subsets of $\{1, 2,...,n\}$ consisting of three elements such that one of them is the arithmetic mean of the other two. [i]Proposed by Walther Janous[/i]

Compute the sum of all positive integers $a\leq 26$ for which there exist integers $b$ and $c$ such that $a+23b+15c-2$ and $2a+5b+14c-8$ are both multiples of $26$.

Given is a cyclic quadrilateral $ABCD$. The point $L_a$ lies in the interior of $BCD$ and is such that its distances to the sides of this triangle are proportional to the lengths of corresponding sides. The points $L_b, L_c$, and $L_d$ are defined analogously. Given that the quadrilateral $L_aL_bL_cL_d$ is cyclic, prove that the quadrilateral $ABCD$ has two parallel sides. (N. Beluhov)

Let $d$ be a natural number. Given two natural numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if the $d$ numbers obtained substituting each one of the digits of $M$ by the digit of $N$ which is on the same position are all multiples of $7$. Find all the values of $d$ for which the following condition is valid: For any two numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if $N$ is a friend of $M$.

A convex pentagon $APBCQ$ is given such that $AB < AC$. The circle $\omega$ centered at point $A{}$ passes through $P{}$ and $Q{}$ and touches the segment $BC$ at point $R{}$. Let the circle $\Gamma$ centered at the point $O{}$ be the circumcircle of the triangle $ABC$. It is known that $AO \perp P Q$ and $\angle BQR = \angle CP R$. Prove that the tangents at points $P{}$ and $Q{}$ to the circle $\omega$ intersect on $\Gamma$.

Find the sum of all positive integers $x$ such that there is a positive integer $y$ satisfying $9x^2 - 4y^2 = 2021$.

Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$. A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$. A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$. Now put a pawn on $(0, 0)$. You can make a (nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B. Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.

Line $l:ax+by+c=0$, where $a,b,c\in\{-3,-2,-1,0,1,2,3\}$, and $a,b,c$ are different. If the bank angle of $l$ is an acute angle, then the number of such lines is________.

Angeline starts with a 6-digit number and she moves the last digit to the front. For example, if she originally had $100823$ she ends up with $310082$. Given that her new number is $4$ times her original number, find the smallest possible value of her original number. [i]Proposed by Angeline Zhao[/i]

Peter is randomly filling boxes with candy. If he has 10 pieces of candy and 5 boxes in a row labeled A, B, C, D, and E, how many ways can he distribute the candy so that no two adjacent boxes are empty?

Let $f$ be a bijection from $\mathbb N$ to itself. Prove that one can always find three natural number $a,b,c$ such that $a<b<c$ and $f(a)+f(c)=2f(b)$.

Given the true statements: (1) If $ a$ is greater than $ b$, then $ c$ is greater than $ d$ (2) If $ c$ is less than $ d$, then $ e$ is greater than $ f$. A valid conclusion is: $ \textbf{(A)}\ \text{If }{a}\text{ is less than }{b}\text{, then }{e}\text{ is greater than }{f}\qquad \\ \textbf{(B)}\ \text{If }{e}\text{ is greater than }{f}\text{, then }{a}\text{ is less than }{b}\qquad \\ \textbf{(C)}\ \text{If }{e}\text{ is less than }{f}\text{, then }{a}\text{ is greater than }{b}\qquad \\ \textbf{(D)}\ \text{If }{a}\text{ is greater than }{b}\text{, then }{e}\text{ is less than }{f}\qquad \\ \textbf{(E)}\ \text{none of these}$

Let $ABC$ be an acute triangle. Let $D$ be the reflection of point $B$ with respect to the line $AC$. Let $E$ be the reflection of point $C$ with respect to the line $AB$. Let $\Gamma_1$ be the circle that passes through $A, B$, and $D$. Let $\Gamma_2$ be the circle that passes through $A, C$, and $E$. Let $P$ be the intersection of $\Gamma_1$ and $\Gamma_2$ , other than $A$. Let $\Gamma$ be the circle that passes through $A, B$, and $C$. Show that the center of $\Gamma$ lies on line $AP$.

The sum of first $n$ items of squence $(a_n)$ : $S_n$ satisfies that $S_n=2a_n-1$, squence $(b_n)$ satisfies that $b_{k+1}=a_k+b_k$ for all $k=1,2,\cdots$. Find the sum of first $n$ items of $(b_n)$.

Let $\omega$ be the circumcircle of triangle $ABC$, and let $D$ be a point on segment $BC$. Let $AD$ intersect $\omega$ at $P$, and let $Q$ lie on minor arc $AC$ of $\omega$ such that $DQ \perp AC$. Given that $CP = CQ$, $\angle DAC=15^{\circ}$, $\angle ADC=120^{\circ}$, and $BD=4$, the value of $CQ$ can be expressed as $a\sqrt{b}-c$ where $a$, $b$, and $c$ are positive integers and $b$ is square-free. Find $a+b+c$. [i]Team #6[/i]

Let $a$ and $b$ be positive real numbers with $a + b =1$. Show that $$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \ge \frac{25}{2}.$$

In an acute-angled triangle $ABC$ on the side $AC$, point $P$ is chosen in such a way that $2AP = BC$. Points $X$ and $Y$ are symmetric to $P$ with respect to vertices $A$ and $C$, respectively. It turned out that $BX = BY$. Find $\angle BCA$.

A triangle $ABC$ with $AB<AC<BC$ is given. The point $P$ is the center of an excircle touching the line segment $AB$ at $D$. The point $Q$ is the center of an excircle touching the line segment $AC$ at $E$. The circumcircle of the triangle $ADE$ intersects $\overline{PE}$ and $\overline{QD}$ again at $G$ and $H$ respectively. The line perpendicular to $\overline{AG}$ at $G$ intersects the side $AB$ at $R$. The line perpendicular to $\overline{AH}$ at $H$ intersects the side $AC$ at $S$. Prove that $\overline{DE}$ and $\overline{RS}$ are parallel.