[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.

Compute the surface area of a rectangular prism with side lengths $2, 3, 4$.

$p$ is an odd prime. Show that $p$ divides $n(n+1)(n+2)(n+3) + 1$ for some integer $n$ iff $p$ divides $m^2 - 5$ for some integer $m$.

Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one side along a side of $T.$ Then inscribe a rectangle $S$ in the triangle formed by the side of $R$ opposite the side on the boundary of $T,$ and the other two sides of $T,$ with one side along the side of $R.$ For any polygon $X,$ let $A(X)$ denote the area of $X.$ Find the maximum value, or show that no maximum exists, of $\tfrac{A(R)+A(S)}{A(T)},$ where $T$ ranges over all triangles and $R,S$ over all rectangles as above.

Find all polynomials $p(x)$ such that for all reals $a, b$ and $c$, with $a+b+c=0$, satisfies $$p(a^3)+p(b^3)+p(c^3)=3p(abc)$$

On a plane, there are $7$ seats. Each is assigned to a passenger. The passengers walk on the plane one at a time. The first passenger sits in the wrong seat (someone else's). For all the following people, they either sit in their assigned seat, or if it is full, randomly pick another. You are the last person to board the plane. What is the probability that you sit in your own seat?