Find all pairs $(x, y)$ of natural numbers $x$ and $y$ such that $\frac{xy^2}{x+y}$ is a prime

A triangle is given on a sphere of radius $1$, the sides of which are arcs of three different circles of radius $1$ centered in the center of a sphere having less than $\pi$ in length and an area equal to a quarter of the area of the sphere. Prove that four copies of such a triangle can cover the entire sphere. A. Zaslavsky

We are given a finite, simple, non-directed graph. Ann writes positive real numbers on each edge of the graph such that for all vertices the following is true: the sum of the numbers written on the edges incident to a given vertex is less than one. Bob wants to write non-negative real numbers on the vertices in the following way: if the number written at vertex $v$ is $v_0$, and Ann's numbers on the edges incident to $v$ are $e_1,e_2,\ldots,e_k$, and the numbers on the other endpoints of these edges are $v_1,v_2,\ldots,v_k$, then $v_0=\sum_{i=1}^k e_iv_i+2022$. Prove that Bob can always number the vertices in this way regardless of the graph and the numbers chosen by Ann. Proposed by [i]Boldizsár Varga[/i], Verőce

Hermione and Ron play a game that starts with $129$ hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven. A player loses if after his or her turn there are two animals of the same species right next to each other. Hermione goes first. Who loses?

Let $f:\mathbb{R} \to\mathbb{R}$ be a function. For every $a\in\mathbb{Z}$ consider the function $f_a : \mathbb{R} \to\mathbb{R}$, $f_a(x) = (x - a)f(x)$. Prove that if there exist infinitely many values $a\in\mathbb{Z}$ for which the functions $f_a$ are increasing, then the function $f$ is monotonic.

Prove that $2^n + 3^n$ is not a perfect cube for any positive integer $n$.

[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$. [url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$ Show that $TB_2AC_2$ is a rectangle. [url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen? [url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$ [url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? $\textbf{(A) } 20 \qquad\textbf{(B) } \dfrac{360}{17} \qquad\textbf{(C) } \dfrac{107}{5} \qquad\textbf{(D) } \dfrac{43}{2} \qquad\textbf{(E) } \dfrac{281}{13} $

The 120 permutations of the AHSME are arranged in dictionary order as if each were an ordinary five-letter word. The last letter of the 85th word in this list is: $ \textbf{(A)}\ \text{A} \qquad \textbf{(B)}\ \text{H} \qquad \textbf{(C)}\ \text{S} \qquad \textbf{(D)}\ \text{M} \qquad \textbf{(E)}\ \text{E} $

In the trapezoid $ABCD$, both $\angle B$ and $\angle C$ are right angles, and all four sides of the trapezoid are tangent to the same circle. If $\overline{AB} = x$ and $\overline{CD} = y$, find the area of $ABCD$ (with proof).

Determine if it is possible to choose nine points in the plane such that there are $n=10$ lines in the plane each of which passes through exactly three of the chosen points. What if $n=11$?

Reconstruct the triangle$ ABC$, in which $\angle B - \angle C = 90^o$ , by the orthocenter $H$ and points $M_1$ and $L_1$ the feet of the median and angle bisector drawn from vertex $A$, respectively. (Gryhoriy Filippovskyi)

Find the coefficient of $x^2$ after expansion and collecting the terms of the following expression (there are $k$ pairs of parentheses): $$((... (((x - 2)^2 - 2)^2 -2)^2 -... -2)^2 - 2)^2$$

A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold: [list=1] [*] each triangle from $T$ is inscribed in $\omega$; [*] no two triangles from $T$ have a common interior point. [/list] Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

Prove that$$p (n)= 2+ \left (p (1) + \cdots + p\left ( \left [\frac {n}{2} \right ] + \chi_1 (n)\right ) + \left (p'_2(n) + \cdots + p' _{ \left [\frac {n}{2} \right ] - 1}(n)\right )\right )$$for every $n \in \mathbb {N}$ with $n>2$ where $\chi $ denotes the principal character Dirichlet modulo 2, i.e.$$ \chi _1 (n) = \begin{cases} 1 & \text{if } (n,2)=1 \\ 0 &\text{if } (n,2)>1 \end{cases} $$with $p (n) $ we denote number of possible partitions of $n $ and $p' _m(n) $ we denote the number of partitions of $n$ in exactly $m$ sumands.

In a board $8$x$8$, the unit squares have numbers $1-64$, as shown. The unit square with a multiple of $3$ on it are red. Find the minimum number of chess' bishops that we need to put on the board such that any red unit square either has a bishop on it or is attacked by at least one bishop. Note: A bishops moves diagonally. [img]https://i.imgur.com/03baBwp.jpeg[/img]

Let $0 \le a$, $b \le 1$ be real numbers. Prove the following inequality: \[\sqrt{a^3b^3}+ \sqrt{(1-a^2)(1-ab)(1-b^2)} \le 1.\] [i](41th Austrian Mathematical Olympiad, regional competition, problem 1)[/i]

A space figure is consisted of $n$ vertexes and $l$ lines connecting these vertices, where $n=q^2+q+1, l\geq\frac{1}{2}q(q+1)^2+1,q\geq2,q\in\mathbb{Z}_+$. The figure satisfies: every four vertices are not coplane, every vertex is connected by at least one line, and there is a vertex connected by at least $q+2$ lines. Prove that there exists a space quadrilateral in the figure. Note: a space quadrilateral is figure with four vertices $A, B, C, D$ and four lines $ AB, BC, CD, DA$.

Determine all functions $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that $f(a) + 2ab + 2f(b)$ divides $f(a)^2 + 4f(b)^2$ for any positive integers $a$ and $b$.

Let $a, b, c, d, e$ and $f$ be some numbers, and $ a \cdot c \cdot e \ne 0$.It is known that the values of the expressions $|ax+b|+|cx+d| $and $|ex+f|$ equal at all values of $x$. Prove that $ad = bc$.

There are no integers $ a,b,c $ that satisfy $ \left( a+b\sqrt{-3}\right)^{17}=c+\sqrt{-3} . $ [i]Dorin Andrica, Mihai Piticari[/i]

Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.

Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\tfrac{2}{9}$, $1$ unit right with probability $\tfrac{2}{9}$, $1$ unit up and $1$ unit right with probability $\tfrac{4}{9}$, and he doesn’t move with probability $\tfrac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A\cdot B$?

Let $ABC$ be a an acute triangle. Points $A_1, B_1$ and $C_1$ are respectively the projections of the vertices $A, B$ and $C$ on the opposite sides of the triangle, the point $H$ is the orthocenter of the triangle, and the point $P$ is the middle of the segment $[AH]$. The lines $BH$ and $A_1C_1$, $P B_1$ and $AB$ intersect respectively at the points $M$ and $N$. Prove that the lines $MN$ and $BC$ are perpendicular.

Let $a,b,c\in\mathbb{Z^{+}}$ such that $$(a^2-1, b^2-1, c^2-1)=1$$ Prove that $$(ab+c, bc+a, ca+b)=(a,b,c)$$ (As usual, $(x,y,z)$ means the greatest common divisor of numbers $x,y,z$) [I]Proposed by A. Golovanov[/i]