Find all functions $f : R^+ \to R^+$ such that $x^2(f(x)+ f(y)) = (x+y)f (f(x)y)$ for any $x,y > 0$.

Let $ABC$ be a triangle with $AB = 3$, $BC = 5$, $AC = 7$, and let $ P$ be a point in its interior. If $G_A$, $G_B$, $G_C$ are the centroids of $\vartriangle PBC$, $\vartriangle PAC$, $\vartriangle PAB$, respectively, find the maximum possible area of $\vartriangle G_AG_BG_C$.

Find the sum of all positive integers $n$ for which \[\frac{15\cdot n!^2+1}{2n-3}\] is an integer. [i]Proposed by Andrew Gu.[/i]

For a triangle $T$ and a line $d$, if the feet of perpendicular lines from a point in the plane to the edges of $T$ all lie on $d$, say $d$ focuses $T$. If the set of lines focusing $T_1$ and the set of lines focusing $T_2$ are the same, say $T_1$ and $T_2$ are equivalent. Prove that, for any triangle in the plane, there exists exactly one equilateral triangle which is equivalent to it.

$1.$ A man walks a certain distance and rides back in $3\frac{3}{4};$ he could ride both ways in $2\frac{1}{2}$ hours. How many hours would it take him to walk both ways $?$

Find the sum $$\sum^{2020}_{n=1} \gcd (n^3 -2n^2 +2021,n^2 -3n +3).$$

[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps? [img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img] [b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side. [b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses . (a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$? (b) Is it possible to do it if $k=4$? [b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$. [b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$. [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that [list=1] [*] $A\cap B=\{1\};$ [*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$; [*] each prime number is a divisor of some number in $A$ and also some number in $B$; [*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$. [*] Each set has infinitely many composite numbers. [/list]

What is the correct ordering of the three numbers, $10^8$, $5{}^1{}^2$, and $2{}^2{}^4$? $ \textbf{(A)}\ 2{}^2{}^4<10^8<5{}^1{}^2 $ $ \textbf{(B)}\ 2{}^2{}^4<5{}^1{}^2<10^8 $ $ \textbf{(C)}\ 5{}^1{}^2<2{}^2{}^4<10^8 $ $ \textbf{(D)}\ 10^8<5{}^1{}^2<2{}^2{}^4$ $ \textbf{(E)}\ 10^8<2{}^2{}^4<5{}^1{}^2 $

The polynomial $P(x)=3x^3-2x^2+ax-b$ has roots $\sin^2\theta$, $\cos^2\theta$, and $\sin\theta\cos\theta$ for some angle $\theta$. Find $P(1)$.

Consider a convex quadrilateral $ABCD$. Denote $M, \ N$ the points of tangency of the circle inscribed in $\triangle ABD$ with $AB, \ AD$, respectively and $P, \ Q$ the points of tangency of the circle inscribed in $\triangle CBD$ with the sides $CD, \ CB$, respectively. Assume that the circles inscribed in $\triangle ABD, \ \triangle CBD$ are tangent. Prove that: a) $ABCD$ is circumscriptible. b) $MNPQ$ is cyclic. c) The circles inscribed in $\triangle ABC, \ \triangle ADC$ are tangent.

Consider a quartet of positive numbers $(a,b,c,d)$. In one step, we transform it to $(ab,bc,cd,da)$. Prove that you can never obtain the initial set if neither of $a,b,c,d$ is $1$.

Find all real numbers $\alpha$ such that both values $\cot(\alpha)$ and $\cot(2\alpha)$ are integers.

Given positive integer $n,k$ such that $2 \le n <2^k$. Prove that there exist a subset $A$ of $\{0,1,\cdots,n\}$ such that for any $x \neq y \in A$, ${y\choose x}$ is even, and $$|A| \ge \frac{{k\choose \lfloor \frac{k}{2} \rfloor}}{2^k} \cdot (n+1)$$

Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.

Each side of a triangle is extended in the same clockwise direction by the length of the given side as shown in the figure. How many times the area of the triangle, obtained by connecting the endpoints, is the area of the original triangle? [img]https://cdn.artofproblemsolving.com/attachments/1/c/a169d3ab99a894667caafee6dbf397632e57e0.png[/img]

Determine all polynomials $P(x)$ such that $P(x^2+1)=(P(x))^2+1$ and $P(0)=0.$

The reals $a, b$ satisfy $$\begin{cases} a^3 - 3a^2 + 5a - 17 = 0 \\ b^3 - 3b^2 + 5b + 11 = 0 .\end{cases}$$ Find $a+b$.

A point $ P$ is randomly selected from the rectangular region with vertices $ (0, 0)$, $ (2, 0)$, $ (2, 1)$, $ (0, 1)$. What is the probability that $ P$ is closer to the origin than it is to the point $ (3, 1)$? $ \textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{2}{3} \qquad \textbf{(C)}\ \frac{3}{4} \qquad \textbf{(D)}\ \frac{4}{5} \qquad \textbf{(E)}\ 1$

Let $XY$ be a segment, which is a diameter of a semi-circle. Let $Z$ be a point on $XY$ and 9 rays from $Z$ are drawn that divide $\angle XZY=180^{\circ}$ into $10$ equal angles. These rays meet the semi-circle at $A_1, A_2, \ldots, A_9$ in this order in the direction from $X$ to $Y$. Prove that the sum of the areas of triangles $ZA_2A_3$ and $ZA_7A_8$ equals the area of the quadrilateral $A_2A_3A_7A_8$.

$65$ distinct natural numbers not exceeding $2016$ are given. Prove that among these numbers we can find four $a,b,c,d$ such that $a+b-c-d$ is divisible by $2016.$

For positive integer $n \ge 3$, find the number of ordered pairs $(a_1, a_2, ... , a_n)$ of integers that satisfy the following two conditions [list=disc] [*]For positive integer $i$ such that $1\le i \le n$, $1 \le a_i \le i$ [*]For positive integers $i,j,k$ such that $1\le i < j < k \le n$, if $a_i = a_j$ then $a_j \ge a_k$ [/list]

The plane is divided into equal squares by parallel lines; i.e., a square net is given. Let $M$ be an arbitrary set of $n$ squares of this net. Prove that it is possible to choose no fewer than $n/4$ squares of $M$ in such a way that no two of them have a common point.

Find all natural numbers $n$ for which equality holds $n + d (n) + d (d (n)) +... = 2021$, where $d (0) = d (1) = 0$ and for $k> 1$, $ d (k)$ is the [i]superdivisor [/i] of the number $k$ (i.e. its largest divisor of $d$ with property $d <k$). (Tomáš Bárta)

Let $n$ be a natural composite number. For each proper divisor $d$ of $n$ we write the number $d + 1$ on the board. Determine all natural numbers $n$ for which the numbers written on the board are all the proper divisors of a natural number $m$. (The proper divisors of a natural number $a> 1$ are the positive divisors of $a$ different from $1$ and $a$.)