Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy \[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\]

Let $ABC$ be a triangle with incenter $I$. Line $AI$ intersects circumcircle of $ABC$ in points $A$ and $D$, $(A \neq D)$. Incircle of $ABC$ touches side $BC$ in point $E$ . Line $DE$ intersects circumcircle of $ABC$ in points $D$ and $F$, $(D \neq F)$. Prove that $\angle AFI = 90^{\circ}$

Consider $A\in \mathcal{M}_{2020}(\mathbb{C})$ such that $$ (1)\begin{cases} A+A^{\times} =I_{2020},\\ A\cdot A^{\times} =I_{2020},\\ \end{cases} $$ where $A^{\times}$ is the adjugate matrix of $A$, i.e., the matrix whose elements are $a_{ij}=(-1)^{i+j}d_{ji}$, where $d_{ji}$ is the determinant obtained from $A$, eliminating the line $j$ and the column $i$. Find the maximum number of matrices verifying $(1)$ such that any two of them are not similar.

Smallville is populated by unmarried men and women, some of them are acquainted. Two city’s matchmakers are aware of all acquaintances. Once, one of matchmakers claimed: “I could arrange that every brunette man would marry a woman he was acquainted with”. The other matchmaker claimed “I could arrange that every blonde woman would marry a man she was acquainted with”. An amateur mathematician overheard their conversation and said “Then both arrangements could be done at the same time! ” Is he right?

An isosceles right triangle is inscribed in a circle of radius 5, thereby separating the circle into four regions. Compute the sum of the areas of the two smallest regions.

Let $ABCD$ be a convex quadrilateral. Extend line $CD$ past $D$ to meet line $AB$ at $P$ and extend line $CB$ past $B$ to meet line $AD$ at $Q$. Suppose that line $AC$ bisects $\angle BAD$. If $AD = \frac{7}{4}$, $AP = \frac{21}{2}$, and $AB = \frac{14}{11}$ , compute $AQ$.

Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity \[ (1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}. \] Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \dots, b_n$ and $n$ (but independent of $a_1, a_2, \dots, a_n$).

The yearly changes in the population census of a town for four consecutive years are, respectively, $25\%$ increase, $25\%$ increase, $25\%$ decrease, $25\%$ decrease. The net change over the four years, to the nearest percent, is: ${{ \textbf{(A)}\ -12 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1}\qquad\textbf{(E)}\ 12} $

Let $\omega$ be the circumcircle of a triangle $ABC$ ($AB>AC$), $E$ be the midpoint of the arc $AC$ which does not contain point $B,$ аnd $F$ the midpoint of the arc $AB$ which does not contain point $C.$ Lines $AF$ and $BE$ meet at point $P,$ line $CF$ and $AE$ meet at point $R,$ and the tangent to $\omega$ at point $A$ meets line $BC$ at point $Q.$ Prove that points $P,Q,R$ are collinear. (M. Kurskiy)

Prove that there are infinitely many sets of four positive integers so that the sum of the squares of any three elements is a perfect square.

The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation. Find the smallest possible number of independent deputies after the vacation.

The student drew a right triangle $ABC$ on the board with a right angle at the vertex $B$ and inscribed in it an equilateral triangle $KMP$ such that the points $K, M, P$ lie on the sides $AB, BC, AC$, respectively, and $KM \parallel AC$. Then the picture was erased, leaving only points $A, P$ and $C$. Restore erased points and lines.

Let $T$ be the smallest positive integers which, when divided by $11,13,15$ leaves remainders in the sets {$7,8,9$}, {$1,2,3$}, {$4,5,6$} respectively. What is the sum of the squares of the digits of $T$ ?

Determine all real numbers $x$ which satisfy the inequality: \[ \sqrt{3-x}-\sqrt{x+1}>\dfrac{1}{2} \]

Solve the system: $$\begin{cases} \dfrac{2x_1^2}{1+x_1^2}=x_2 \\ \\ \dfrac{2x_2^2}{1+x_2^2}=x_3\\ \\ \dfrac{2x_3^2}{1+x_3^2}=x_1\end{cases}$$

For every integer $m\ge 1$, let $\mathbb{Z}/m\mathbb{Z}$ denote the set of integers modulo $m$. Let $p$ be a fixed prime and let $a\ge 2$ and $e\ge 1$ be fixed integers. Given a function $f\colon \mathbb{Z}/a\mathbb{Z}\to \mathbb{Z}/p^e\mathbb{Z}$ and an integer $k\ge 0$, the $k$[i]th finite difference[/i], denoted $\Delta^k f$, is the function from $\mathbb{Z}/a\mathbb{Z}$ to $\mathbb{Z}/p^e\mathbb{Z}$ defined recursively by \begin{align*} \Delta^0 f(n)&=f(n)\\ \Delta^k f(n)&=\Delta^{k-1}f(n+1)-\Delta^{k-1}f(n) & \text{for } k=1,2,\dots. \end{align*} Determine the number of functions $f$ such that there exists some $k\ge 1$ for which $\Delta^kf=f$. [i]Holden Mui[/i]

Let $S$ be the set of all numbers which are the sum of the squares of three consecutive integers. Then we can say that: $\textbf{(A) }\text{No member of }S\text{ is divisible by }2\qquad$ $\textbf{(B) }\text{No member of }S\text{ is divisible by }3\text{ but some member is divisible by }11\qquad$ $\textbf{(C) }\text{No member of }S\text{ is divisible by }3\text{ or }5\qquad$ $\textbf{(D) }\text{No member of }S\text{ is divisible by }3\text{ or }7\qquad$ $\textbf{(E) }\text{None of these}$

Let $f:[0,1] \to \mathbb{R}$ be a continuous function with $f(1)=0.$ Prove that the limit $$\lim_{t \nearrow 1} \left( \frac{1}{1-t} \int\limits_0^1x(f(tx)-f(x)) \mathrm{d}x\right)$$ exists and find its value.

[b]10.[/b] A car is used by $n$ drivers. Every morning the drivers choose by drawing that one of them who will drive the car that day. Let us define the random variable $\mu (n)$ as the least positive integer such that each driver drives at least one day during the first $\mu (n)$ days. Find the limit distribution of the random variable $\frac {\mu (n) -n \log n}{n}$ as $n \to \infty$. [b](P. 9)[/b]

Find all rational terms of sequence defined by formula $ a_n=\sqrt{\frac{9n-2}{n+1}}, n \in N $

$n$ is a given positive integer, such that it’s possible to weigh out the mass of any product weighing $1,2,3,\cdots ,ng$ with a counter balance without sliding poise and $k$ counterweights, which weigh $x_ig(i=1,2,\cdots ,k),$ respectively, where $x_i\in \mathbb{N}^*$ for any $i \in \{ 1,2,\cdots ,k\}$ and $x_1\leq x_2\leq\cdots \leq x_k.$ $(1)$Let $f(n)$ be the least possible number of $k$. Find $f(n)$ in terms of $n.$ $(2)$Find all possible number of $n,$ such that sequence $x_1,x_2,\cdots ,x_{f(n)}$ is uniquely determined.

If ${\cos^5 x}-{\sin^5 x}<7({\sin^3 x}-{\cos ^3 x}) $ (for $x\in [ 0,2\pi) $), then find the range of $x$.

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

Given are two sets of positive numbers with the same sum. The first set has $m$ numbers and the second $n$. Prove that you can find a set of less than $m+n$ positive numbers which can be arranged to part fill an $m \times n$ array, so that the row and column sums are the two given sets.

Let $ (\log_2(x))^2 \minus{} 4 \cdot \log_2(x) \minus{} m^2 \minus{} 2m \minus{} 13 \equal{} 0$ be an equation in $ x.$ Prove: [b](a)[/b] For any real value of $ m$ the equation has two distinct solutions. [b](b)[/b] The product of the solutions of the equation does not depend on $ m.$ [b](c)[/b] One of the solutions of the equation is less than 1, while the other solution is greater than 1. Find the minimum value of the larger solution and the maximum value of the smaller solution.