Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.

A polygon with $1997$ vertices is given. Write a positive real number each vertex such that, each number equal to its right and left numbers' arithmetic or geometric mean. Prove that all numbers are equal.

In how many ways can $100$ be written as the sum of three positive integers $x, y$, and $z$ satisfying $x < y < z$ ?

Let $M$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$. Let $P$ and $Q$ be the centroids of triangles $AMD$ and $BMC$ respectively. Let $R$ and $S$ are the orthocenters of triangles $AMB$ and $CMD$. Prove that the lines $P Q$ and $RS$ are perpendicular to each other.

A circle touches the extensions of sides $CA$ and $CB$ of triangle $ABC$, and also touches side $AB$ of this triangle at point $P$. Prove that the radius of the circle tangent to segments $AP$, $CP$ and the circumscribed circle of this triangle is equal to the radius of the inscribed circle in this triangle.

For a function $ f(x)\equal{}(\ln x)^2\plus{}2\ln x$, let $ C$ be the curve $ y\equal{}f(x)$. Denote $ A(a,\ f(a)),\ B(b,\ f(b))\ (a<b)$ the points of tangency of two tangents drawn from the origin $ O$ to $ C$ and the curve $ C$. Answer the following questions. (1) Examine the increase and decrease, extremal value and inflection point , then draw the approximate garph of the curve $ C$. (2) Find the values of $ a,\ b$. (3) Find the volume by a rotation of the figure bounded by the part from the point $ A$ to the point $ B$ and line segments $ OA,\ OB$ around the $ y$-axis.

Consider $\left(\triangle_n=A_nB_nC_n\right)_{n\ge 1}$. We define points $A_n',B_n',C_n'$ in sides $C_nB_n,A_nC_n,B_nA_n$ such that $$(n+1)B_nA_n'=C_nA_n',~(n+1)C_nB_n'=A_nB_n',~(n+1)A_nC_n'=B_nC_n'$$$\triangle_{n+1}$ is defined by the intersections of $A_nA_n',B_nB_n',C_nC_n'$. If $S_n$ denotes the area of $\triangle_n$. Find $\frac{S_1}{S_{2020}}$. [i]Proposed by Alejandro Madrid, Valle[/i]

Alina knows how to twist a periodic decimal fraction in the following way: she finds the minimum preperiod of the fraction, then takes the number that makes up the period and rearranges the last one in it digit to the beginning of the number. For example, from the fraction, $0.123(56708)$ she will get $0.123(85670)$. What fraction will Alina get from fraction $\frac{503}{2022}$ ?

We wish to color the cells of a $n \times n$ chessboard with $k$ different colors such that for every $i\in \{1,2,...,n\}$, the $2n-1$ cells on $i$. row and $i$. column have all different colors. a) Prove that for $n=2001$ and $k=4001$, such coloring is not possible. b) Show that for $n=2^{m}-1$ and $k=2^{m+1}-1$, such coloring is possible.

How many integers $n$ are there such that $n^3+8$ has at most $3$ positive divisors? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None of the above} $

There is a black token in the lower-left corner of a board $m \times n$ ($m, n \ge 3$), and there are white tokens in the lower-right and upper-left corners of this board. Petryk and Vasyl are playing a game, with Petryk playing with a black token and Vasyl with white tokens. Petryk moves first. In his move, a player can perform the following operation at most two times: choose any his token and move it to any adjacent by side cell, with one restriction: you can't move a token to a cell where at some point was one of the opponents' tokens. Vasyl wins if at some point of the game white tokens are in the same cell. For which values of $m, n$ can Petryk prevent him from winning? [i](Proposed by Arsenii Nikolaiev)[/i]

Let $n \ge 2$ be a positive integer and let $A \in \mathcal{M}_n(\mathbb{R})$ be a matrix such that $A^2=-I_n$. If $B \in \mathcal{M}_n(\mathbb{R})$ and $AB = BA$, prove that $\det B \ge 0$.

Consider the function $f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying \[f_k(x)=x+k\varphi(x)\] where $\varphi(x)$ is Euler's totient function, that is, the number of positive integers up to $x$ coprime to $x$. We define a sequence $a_1,a_2,...,a_{10}$ with [list] [*] $a_1=c$, and [*] $a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10$ [/list] Is it possible to choose the initial value $c\ne 1$ such that each term is a multiple of the previous, if (a) $k=2025$ ? (b) $k=2065$ ? [i]Proposed by chorn[/i]

If $x$, $y$, and $z$ are distinct positive integers such that $x^2+y^2=z^3$, what is the smallest possible value of $x+y+z$?

There are pairs of square numbers with the following two properties: (1) Their decimal representations have the same number of digits, with the first digit starting is different from $0$ . (2) If one appends the second to the decimal representation of the first, the decimal representation results another square number. Example: $16$ and $81$; $1681 = 41^2$. Prove that there are infinitely many pairs of squares with these properties.

Consider a Riemannian metric on the vector space ${\Bbb{R}^n}$ which satisfies the property that for each two points ${a,b}$ there is a single distance minimising geodesic segment ${g(a,b)}$. Suppose that for all ${a \in \Bbb{R}^n}$, the Riemannian distance with respect to ${a}, {\rho_a : \Bbb{R}^n \rightarrow \Bbb{R}}$ is convex and differentiable outside of ${a}$. Prove that if for a point ${x \neq a,b}$ we have \[ \displaystyle \partial_i \rho_a(x)=-\partial_i \rho_b(x),\ i=1,\cdots, n\] then ${x}$ is a point on ${g(a,b)}$ and conversely. [i]Proposed by Lajos Tamássy and Dávid Kertész[/i]

Let $a_1, a_2,...,a_{51}$ be non-zero elements of a field of characteristic $p$. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_1, ... , b_{51}$. If this new sequence is a permutation of the original one, find all possible values of $p$.

Edward starts in his house, which is at (0,0) and needs to go point (6,4), which is coordinate for his school. However, there is a park that shaped as a square and has coordinates (2,1),(2,3),(4,1), and (4,3). There is no road for him to walk inside the park but there is a road for him to walk around the perimeter of the square. How many different shortest road routes are there from Edward's house to his school?

What is the largest number of points that can exist on a plane so that each distance between any two of them is an odd integer?

Compute $\left\lfloor \sum_{k=0}^{10}\left(3+2\cos\left(\frac{2k\pi}{11}\right)\right)^{10}\right\rfloor \pmod{100}.$

Let $a, b, c$ be the lengths of the sides of a triangle, and let $S$ be its area. We know that $S = \frac14 (c^2 - a^2 - b^2)$. Prove that $\angle C = 135^o$.

Points $M$ and $N$ are the midpoints of sides $AB$ and $CD$, respectively of quadrilateral $ABCD$. It is known that $BC // AD$ and $AN = CM$. Is it true that $ABCD$ is parallelogram?

Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

$ 30$ students participated in the mathematical Olympiad. Each of them was given $ 8$ problems to solve. Jury estimated their work with the following rule: 1) Each problem was worth $ k$ points, if it wasn't solved by exactly $ k$ students; 2) Each student received the maximum possible points in each problem or got $ 0$ in it; Lasha got the least number of points. What's the maximal number of points he could have? Remark: 1) means that if the problem was solved by exactly $ k$ students, than each of them got $ 30 \minus{} k$ points in it.

Find all square-free integers $d$ for which there exist positive integers $x, y$ and $n$ satisfying $x^2+dy^2=2^n$ Submitted by Kada Williams, Cambridge