There is convex $n-$gon. We color all its sides and also diagonals, that goes out from one vertex. So we have $2n-3$ colored segments. We write positive numbers on colored segments. In one move we can take quadrilateral $ABCD$ such, that $AC$ and all sides are colored, then remove $AC$ and color $BD$ with number $\frac{xz+yt}{w}$, where $x,y,z,t,w$ - numbers on $AB,BC,CD,DA,AC$. After some moves we found that all colored segments are same that was at beginning. Prove, that they have same number that was at beginning.

If $f\left(\frac{x}{x - 1}\right) = \frac{1}{x}$ for all $x \ne 0,1$ and $0 < \theta < \frac{\pi}{2}$, then $f(\sec^{2}\theta) =$ $ \textbf{(A) }\sin^{2}\theta\qquad\textbf{(B) }\cos^{2}\theta\qquad\textbf{(C) }\tan^{2}\theta\qquad\textbf{(D) }\cot^{2}\theta\qquad\textbf{(E) }\csc^{2}\theta $

For $a,b,c$ positive reals find the minimum value of \[ \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}. \]

We consider the sequence $t_1,t_2,t_3,...$ By $P_n$ we mean the product of the first $n$ terms of the sequence. Given that $t_{n+1} = t_n \cdot t_{n+2}$ for each $n$, and that $P_{40} = P_{80} = 8$. Calculate $t_1$ and $t_2$.

Fredy starts at the origin of the Euclidean plane. Each minute, Fredy may jump a positive integer distance to another lattice point, provided the jump is not parallel to either axis. Can Fredy reach any given lattice point in 2023 jumps or less? [i]Proposed by Tony Wang[/i]

Find all integer values can take $n$ such that $$\cos(2x)=\cos^nx - \sin^nx$$ for every real number $x$.

$p$ is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers $(x,y)$ that satisfy the following equation. $$p^2 x^4-6px^2+1=y^2$$

A sequence $ \left( x_n\right)_{n\ge 0} $ of real numbers satisfies $ x_0>1=x_{n+1}\left( x_n-\left\lfloor x_n\right\rfloor\right) , $ for each $ n\ge 1. $ Prove that if $ \left( x_n\right)_{n\ge 0} $ is periodic, then $ x_0 $ is a root of a quadratic equation. Study the converse.

The last four digits of a perfect square are equal. Prove that all of them are zeros.

Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$

Let $PA$ and $PB$ be the tangent of a circle $\omega$ from a point $P$ outside the circle. Let $M$ be any point on $AP$ and $N$ is the midpoint of segment $AB$. $MN$ cuts $\omega$ at $C$ such that $N$ is between $M$ and $C$. Suppose $PC$ cuts $\omega$ at $D$ and $ND$ cuts $PB$ at $Q$. Prove $MQ$ is parallel to $AB$.

Compute the number of ordered pairs of integers $(a, b),$ with $2 \le a, b \le 2021,$ that satisfy the equation \[a^{\log_b \left(a^{-4}\right)} = b^{\log_a \left(ba^{-3}\right)}.\]

Let $n$ be a positive integer. Suppose that $A,B,$ and $M$ are $n\times n$ matrices with real entries such that $AM=MB,$ and such that $A$ and $B$ have the same characteristic polynomial. Prove that $\det(A-MX)=\det(B-XM)$ for every $n\times n$ matrix $X$ with real entries.

Find the minimum of $\sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2$ where $x$ is a real numbers

Does there exist a circle and an infinite set of points on it such that the distance between any two points of the set is rational?

Given $2012$ distinct points $A_1,A_2,\dots,A_{2012}$ on the Cartesian plane. For any permutation $B_1,B_2,\dots,B_{2012}$ of $A_1,A_2,\dots,A_{2012}$ define the [i]shadow[/i] of a point $P$ as follows: [i]Point $P$ is rotated by $180^{\circ}$ around $B_1$ resulting $P_1$, point $P_1$ is rotated by $180^{\circ}$ around $B_2$ resulting $P_2$, ..., point $P_{2011}$ is rotated by $180^{\circ}$ around $B_{2012}$ resulting $P_{2012}$. Then, $P_{2012}$ is called the shadow of $P$ with respect to the permutation $B_1,B_2,\dots,B_{2012}$.[/i] Let $N$ be the number of different shadows of $P$ up to all permutations of $A_1,A_2,\dots,A_{2012}$. Determine the maximum value of $N$. [i]Proposer: Hendrata Dharmawan[/i]

Each integer is colored with exactly one of $3$ possible colors -- black, red or white -- satisfying the following two rules : the negative of a black number must be colored white, and the sum of two white numbers (not necessarily distinct) must be colored black. [b](a)[/b] Show that, the negative of a white number must be colored black and the sum of two black numbers must be colored white. [b](b)[/b] Determine all possible colorings of the integers that satisfy these rules.

Define the polynomial $f(x) = x^4 + x^3 + x^2 + x + 1$. Compute the number of positive integers $n$ less than equal to $2022$ such that $f(n)$ is $1$ more than multiple of $5$.

In triangle $ ABC$, $ M$ is the midpoint of side $ BC$ and $ G$ is the centroid of triangle $ ABC$. A line $ l$ passes through $ G$, intersecting line $ AB$ at $ P$ and line $ AC$ at $ Q$, where $ P\ne B$ and $ Q\ne C$. If $ [XYZ]$ denotes the area of triangle $ XYZ$, show that $ \frac{[BGM]}{[PAG]}\plus{}\frac{[CMG]}{[QGA]}\equal{}\frac32$.

Positive reals $a,b,c$ satisfy $ab+bc+ca=abc$. Prove that: $\frac{a^4+b^4}{ab(a^3+b^3)} + \frac{b^4+c^4}{bc(b^3+c^3)}+\frac{c^4+a^4}{ca(c^3+a^3)} \geq 1$

A number is called [i]good[/i] if it can be written as the sum of the squares of three consecutive positive integers. A number is called excellent if it can be written as the sum of the squares of four consecutive positive integers. (For instance, $14 = 1^2 + 2^2 + 3^2$ is good and $30 =1^2 +2^2 +3^2+4^2$ is excellent.) A good number $G$ is called [i]splendid[/i] if there exists an excellent number $E$ such that $3G-E = 2025.$ If the sum of all splendid numbers is $S,$ find the remainder when $S$ is divided by $1000.$

Let $f$ of degree at most 13 such that $f(k) = 13^k$ for $0 \leq k \leq 13$. Compute the last three digits of $f(14)$. [i]Proposed by Kaylee Ji[/i]

Determine all pairs of real numbers $(k, d)$ such that the system of equations $$\begin{cases} x^3 + y^3 = 2 \\ kx + d = y\end{cases}$$ has no solutions $(x, y)$ with $x$ and $y$ real numbers.

Evaluate \[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]

Let positive numbers $a_1, a_2, ..., a_{3n}$ $(n \geq 2)$ constitute an arithmetic progression with common difference $d > 0$. Prove that among any $n + 2$ terms in this progression, there exist two terms $a_i, a_j$ $(i \neq j)$ satisfying $1 < \frac{|a_i - a_j|}{nd} < 2$.