Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals. Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded? [i]Proposed by Mihai Bălună, Romania[/i]

Show that for $0 <\epsilon <1$ the expression $(x+y)^n(x^2-(2-\epsilon)xy+y^2)$ is a polynomial with positive coefficients for $n$ sufficiently large and integral. For $\epsilon =.002$ find the smallest admissible value of $n$.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$ (1) Find the minimum of $f_{2020}$. (2) Find the minimum of $f_{2020} \cdot f_{2021}$.

Let $a,b,c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $ax^2+bx+c, bx^2+cx+a,$ and $cx^2+ax+b $.

On an infinite checkered plane $100$ chips in form of a $10\times 10$ square are given. These chips are rearranged such that any two adjacent (by side) chips are again adjacent, moreover no two chips are in the same cell. Prove that the chips are again in form of a square.

Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$, where $F_n$ is the Fibonacci sequence ($F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$) Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$

$ ABCD$ is convex $ AD\parallel BC$, $ AC\perp BD$. $ M$ is interior point of $ ABCD$ which is not a intersection of diagonals $ AC$ and $ BD$ such that $ \angle AMB \equal{}\angle CMD \equal{}\frac{\pi}{2}$ .$ P$ is intersection of angel bisectors of $ \angle A$ and $ \angle C$. $ Q$ is intersection of angel bisectors of $ \angle B$ and $ \angle D$. Prove that $ \angle PMB \equal{}\angle QMC$.

Let $ABC$ be a triangle without obtuse angles, $M$ the midpoint of $BC$, $K$ the midpoint of $BM$. What is the largest value of the angle $\angle KAM$?

Prove that in a triangle one of the sides is twice as large as the other if and only if a median and an angle bisector of this triangle are perpendicular

Solve the system of equations \[\begin{cases}x^5=y+y^5\\ y^5=z+z^5\\ z^5=t+t^5\\ t^5=x+x^5.\end{cases}\]

The sequence of real numbers $\{a_n\}$, $n \in \mathbb{N}$ satisfies the following condition: $a_{n+1}=a_n(a_n+2)$ for any $n \in \mathbb{N}$. Find all possible values for $a_{2004}$.

Let $ f:\mathbb{R}\longrightarrow (0,\infty ) $ be a continuous and periodic function having a period of $ 2, $ and such that the integral $ \int_0^2 \frac{f(1+x)}{f(x)} dx $ exists. Show that $$ \int_0^2 \frac{f(1+x)}{f(x)} dx\ge 2, $$ with equality if and only if $ 1 $ is also a period of $ f. $

Let $ABCD$ be a quadrilateral inscribed in the unit circle such that $\angle BAD$ is $30$ degrees. Let $m$ denote the minimum value of $CP + PQ + CQ$, where $P$ and $Q$ may be any points lying along rays $AB$ and $AD$, respectively. Determine the maximum value of $m$.

For a positive integer $n$, an [i]$n$-branch[/i] $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to [i]appear[/i] in $B$ if it is an element of the last set $S_m$. Define an [i]$n$-plant[/i] to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it [i]perfect[/i] if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches. Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Yang Liu[/i]

Points $K$ and $L$ are chosen on the sides $AB$ and $BC$ of the isosceles $\triangle ABC$ ($AB = BC$) so that $AK +LC = KL$. A line parallel to $BC$ is drawn through midpoint $M$ of the segment $KL$, intersecting side $AC$ at point $N$. Find the value of $\angle KNL$.

Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$.

Three roots of the quartic polynomial $f(x)=x^4+ax^3+bx+c$ are $-1$, $3$, and $5$. What is $a+b-c$?

A square has $2$ diagonals. A regular pentagon has $5$ diagonals. $n$ is the smallest positive integer such that a regular $n$-gon has greater than or equal to $2024$ diagonals. What is the sum of the digits of $n$? \[\mathrm a. ~10\qquad \mathrm b. ~11 \qquad \mathrm c. ~12 \qquad\mathrm d. ~13\qquad\mathrm e. ~14\]

Let $ABCD$ be a cyclic quadrilateral with circumcentre $O$ and with $AC$ perpendicular to $BD$. Points $X$ and $Y$ lie on the circumcircle of the triangle $BOD$ such that $\angle AXO=\angle CYO=90^{\circ}$. Let $M$ be the midpoint of $AC$. Prove that $BD$ is tangent to the circumcircle of the triangle $MXY$.

Let $n$ be a positive integer with $k\ge22$ divisors $1=d_{1}< d_{2}< \cdots < d_{k}=n$, all different. Determine all $n$ such that \[{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.\]

Alina and Bogdan play a game on a $2\times n$ rectangular grid ($n\ge 2$) whose sides of length $2$ are glued together to form a cylinder. Alternating moves, each player cuts out a unit square of the grid. A player loses if his/her move causes the grid to lose circular connection (two unit squares that only touch at a corner are considered to be disconnected). Suppose Alina makes the first move. Which player has a winning strategy?

Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.

Positive real numbers $a,b,c$ satisfy $$2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2.$$ Prove that $$2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2 \geq(ab+bc+ca)^2.$$

$\triangle ABC$ has $AB=4$ and $AC=6$. Let point $D$ be on line $AB$ so that $A$ is between $B$ and $D$. Let the angle bisector of $\angle BAC$ intersect line $BC$ at $E$, and let the angle bisector of $\angle DAC$ intersect line $BC$ at $F$. Given that $AE=AF$, find the square of the circumcircle's radius' length.