Given regular $17$-gon $A_1 ... A_{17}$. Prove that two triangles formed by lines $A_1A_4, A_2A_{10}, A_{13}A_{14}$ and $A_2A_3, A_4A_6 A_{14}A_{15} $ are equal. (N.Beluhov)

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

For any measurable set $H \subset R$ , we define the sequence $a_n(H)$ by the formula: $$a_n(H) = \lambda \bigg([0,1] \setminus \bigcup_{k = n}^{2n} (H + \log_2 k) \bigg)$$ where $\lambda$ denotes the Lebesgue measure and $\log_2$ denotes the binary logarithm. Prove that there is a measurable, 1-periodic, positive measure set $H \subset R$ , such that the sequence $a_n( H )$ does not belong to any space $l_p$ ($1 \leq p < \infty$). [hide=not sure about this part]For what numbers $1 \leq p <\infty$ is it true that whenever H is 1-periodic, positive measure, the sequence $a_n( H )$ belongs to the space $l_p$?[/hide]

Find all functions $f : R_{>0} \to R$ such that $f \left(\frac{x}{y}\right) = f(x) + f(y) - f(x)f(y)$ for all $x, y \in R_{>0}$. Here, $R_{>0}$ denotes the set of all positive real numbers. Nguyễn Duy Thái Sơn

There are two letter sequences $A$ and $B$, both with length $100$ letters. In one move you can insert in any place of sequence ( possibly to start or to end) any number of same letters or remove any number of consecutive same letters. Prove that it is possible to make second sequence from first sequence using not more than $100$ moves.

Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$. [i]Author: Stephan Wagner, Austria[/i]

[b]p1.[/b] The sequence $\{x_n\}$ is defined by $$x_{n+1} = \begin{cases} 2x_n - 1, \,\, if \,\, \frac12 \le x_n < 1 \\ 2x_n, \,\, if \,\, 0 \le x_n < \frac12 \end{cases}$$ where $0 \le x_0 < 1$ and $x_7 = x_0$. Find the number of sequences satisfying these conditions. [b]p2.[/b] Let $M = \{1, . . . , 2022\}$. For any nonempty set $X \subseteq M$, let $a_X$ be the sum of the maximum and the minimum number of $X$. Find the average value of $a_X$ across all nonempty subsets $X$ of $M$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)

Let $ a_1$, $ a_2$, $ a_3$, $ a_4$, $ a_5$ be real numbers satisfying the following equations: $ \frac{a_1}{k^2\plus{}1}\plus{}\frac{a_2}{k^2\plus{}2}\plus{}\frac{a_3}{k^2\plus{}3}\plus{}\frac{a_4}{k^2\plus{}4}\plus{}\frac{a_5}{k^2\plus{}5} \equal{} \frac{1}{k^2}$ for $ k \equal{} 1, 2, 3, 4, 5$ Find the value of $ \frac{a_1}{37}\plus{}\frac{a_2}{38}\plus{}\frac{a_3}{39}\plus{}\frac{a_4}{40}\plus{}\frac{a_5}{41}$ (Express the value in a single fraction.)

Find all $4$-tuple of real numbers $(x, y, z, w)$ that satisfy the following system of equations: $$x^2 + y^2 + z^2 + w^2 = 4$$ $$\frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2 }+\frac{1}{w^2} = 5 -\frac{1}{(xyzw)^2}$$

For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$. [i]Proposed by Juhan Aru, Estonia[/i]

We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic.

In a circle with center $ O$, $ AD$ is a diameter, $ ABC$ is a chord, $ BO \equal{} 5$, and $ \angle ABO = \stackrel{\frown}{CD} = 60^{\circ}$. Then the length of $ BC$ is: [asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C); draw(C--A--D^^B--O^^Circle(O,1)); pair point=O; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$O$", O, dir(305)); label("$5$", B--O, dir(O--B)*dir(90)); label("$60^\circ$", dir(185), dir(185)); label("$60^\circ$", B+0.05*dir(-25), dir(-25));[/asy] $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 3 \plus{} \sqrt3 \qquad \textbf{(C)}\ 5 \minus{} \frac{\sqrt3}{2} \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{none of the above}$

The diagonals of the cyclic quadrilateral $ABCD$ intersect at the point $E$. Let $P$ and $Q$ are the centers of the circles circumscribed around the triangles $BCE$ and $DCE$, respectively. A straight line passing through the point $P$ parallel to $AB$, and a straight line passing through the point $Q$ parallel to $AD$, intersect at the point $R$. Prove that the point $R$ lies on segment $AC$.

Let $n\ge 2,$ be a positive integer. Numbers $\{1,2,3, ...,n\}$ are written in a row in an arbitrary order. Determine the smalles positive integer $k$ with the property: everytime it is possible to delete $k$ numbers from those written on the table, such that the remained numbers are either in an increasing or decreasing order.

[b]a)[/b] Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of real numbers having the property that $ \left| x_{n+1} -x_n \right|\leqslant 1/2^n, $ for any $ n\geqslant 1. $ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent. [b]b)[/b] Create a sequence $ \left( y_n \right)_{n\ge 1} $ of real numbers that has the following properties: $ \text{(i) } \lim_{n\to\infty } \left( y_{n+1} -y_n \right) = 0 $ $ \text{(ii) } $ is bounded $ \text{(iii) } $ is divergent [i]Eugen Popa[/i]

Sam has 1 Among Us task left. He and his task are located at two randomly chosen distinct vertices of a 2021-dimensional unit hypercube. Let $E$ denote the expected distance he has to walk to get to his task, given that he is only allowed to walk along edges of the hypercube. Compute $\lceil 10E \rceil$. [i]Proposed by Sammy Charney[/i]

Find the largest value of n, such that there exists a polygon with n sides, 2 adjacent sides of length 1, and all his diagonals have an integer length.

$ \Delta ABC$ is a triangle such that $ AB \neq AC$. The incircle of $ \Delta ABC$ touches $ BC, CA, AB$ at $ D, E, F$ respectively. $ H$ is a point on the segment $ EF$ such that $ DH \bot EF$. Suppose $ AH \bot BC$, prove that $ H$ is the orthocentre of $ \Delta ABC$. Remark: the original question has missed the condition $ AB \neq AC$

On an acute triangle $ABC$ we draw the internal bisector of $<ABC$, $BE$, and the altitude $AD$, ($D$ on $BC$), show that $<CDE$ it's bigger than 45 degrees.

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

The angles of a convex quadrilateral form an arithmetic sequence in clockwise order, and its side lengths also form an arithmetic sequence (but not necessarily in clockwise order). If the quadrilateral is not a square, and its shortest side has length $1$, then its perimeter is $a + \sqrt{b}4$, where $ a$ and $b$ are positive integers. What is the value of $a + b$?

Determine all positive integers $a,b,c$ satisfying $a^{(b^c)}=(b^a)^c$

Two parallel lines $a$ and $b$ meet two other lines $c$ and $d$. Let $A$ and $A'$ be the points of intersection of $a$ with $c$ and $d$, respectively. Let $B$ and $B'$ be the points of intersection of $b$ with $c$ and $d$, respectively. If $X$ is the midpoint of the line segment $A A'$ and $Y$ is the midpoint of the segment $BB'$, prove that $$|XY| \le \frac{|AB|+|A'B'|}{2}.$$

If $x=\dfrac{1-i\sqrt{3}}{2}$ where $i=\sqrt{-1}$, then $\dfrac{1}{x^2-x}$ is equal to $\textbf{(A) }-2\qquad\textbf{(B) }-1\qquad\textbf{(C) }1+i\sqrt{3}\qquad\textbf{(D) }1\qquad \textbf{(E) }2$