Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$

Show that there do not exist non-zero real numbers $a,b,c$ such that the following statements hold simultaneously: $\bullet$ the equation $ax^2 + bx + c = 0$ has two distinct roots $x_1,x_2$; $\bullet$ the equation $bx^2 + cx + a = 0$ has two distinct roots $x_2,x_3$; $\bullet$ the equation $cx^2 + ax + b = 0$ has two distinct roots $x_3,x_1$. (Note that $x_1,x_2,x_3$ may be real or complex numbers.)

Kelvin the Frog writes 2020 words on a blackboard, with each word chosen uniformly randomly from the set $\{\verb|happy|, \verb|boom|, \verb|swamp|\}$. A multiset of seven words is [i]merry[/i] if its elements can spell $``\verb|happy happy boom boom swamp swamp swamp|."$ For example, the eight words \[\verb|swamp|, \verb|happy|, \verb|boom|, \verb|swamp|, \verb|swamp|, \verb|boom|, \verb|swamp|, \verb|happy|\] contain four merry multisets. Determine the expected number of merry multisets contained in the words on the blackboard. [size=6][url]http://www.hpmor.com/chapter/12[/url][/size]

The cities of Terra Brasilis are connected by some roads. There are no two cities directly connected by more than one road. It is known that it is possible to go from one city to any other using one or more roads. We call [i]role[/i] any closed road route (ie, it starts in a city and ends in the same city) that does not pass through a city more than once. At Terra Brasilis, all roles go through an odd number of cities. The government of Terra Brasilis decided to close some roles for reform. When you close a role, all it;s roads are closed, so traffic is not allowed on these roads. By doing this, Terra Brasilis was divided into several regions such that from any city in each region it is possible to reach any other in the same region by road, but it is not possible to reach cities in other regions. Prove that the number of regions is odd [hide=original wording]As cidades da Terra Brasilis sao conectadas por algumas estradas. Nao ha duas cidades conectadas diretamente por mais de uma estrada. Sabe-se que, e possivel ir de uma cidade para qualquer outra utilizando uma ou mais estradas. Chamamos de rol^e qualquer rota fechada de estradas (isto e, comeca em uma cidade e termina na mesma cidade) que nao passa por uma cidade mais de uma vez. Na Terra Brasilis, todos os roles passam por quantidades impares de cidades. O governo da Terra Brasilis decidiu fechar alguns roles para reforma. Ao fechar um role, todas as suas estradas sao interditadas, de modo que nao e permitido o trafego nessas estradas. Ao fazer isso, a Terra Brasilis ficou dividida em varias regioes de modo que, de qualquer cidade de cada regiao e possivel hegar a qualquer outra da mesma regiao atraves de estradas, mas nao e possivel hegar a cidades de outras regioes. Prove que o numero de regioes e impar.[/hide]

In the triangle $ABC$, ${{A}_{1}}$ and ${{C}_{1}} $ are the midpoints of sides $BC $ and $AB$ respectively. Point $P$ lies inside the triangle. Let $\angle BP {{C}_{1}} = \angle PCA$. Prove that $\angle BP {{A}_{1}} = \angle PAC $.

Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$ Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$ [i]Proposed by Ma Zhao Yu

Prove that there doesn't exist any positive integer $n$ such that $2n^2+1,3n^2+1$ and $6n^2+1$ are perfect squares.

Let $f$ be defined as below for integers $n \ge 0$ and $a_0, a_1, ...$ such that $\sum_{i\ge 0}a_i$ is finite: $$f(n; a_0, a_1, ...) = \begin{cases} a_{2020}, & \text{ $n = 0$} \\ \sum_{i\ge 0} a_i f(n-1;a_0,...,a_{i-1},a_i-1,a_{i+1}+1,a_{i+2},...)/ \sum_{i\ge 0}a_i & \text{$n > 0$} \end{cases}$$. Find the nearest integer to $f(2020^2; 2020, 0, 0, ...)$.

There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?

Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$, for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.

Circle $I$ is the incircle of $\triangle ABC$. Circle $I$ is tangent to sides $BC$ and $AC$ at $M,N$ respectively. $E,F$ are midpoints of sides $AB$ and $AC$ respectively. Lines $EF, BI$ intersect at $D$. Show that $M,N,D$ are collinear.

Let $A$ and $B$ be $2\times 2$ matrices with integer entries such that $A, A+B, A+2B, A+3B,$ and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.

Let $K, L, M$ denote three points on the sides $BC$, $AB$ and $BC$ of $\triangle{ABC}$, so that $ALKM$ is a parallelogram. Points $S$ and $T$ are chosen on lines $KL$ and $KM$ respectively, so that the quadrilaterals $AKBS$ and $AKCT$ are both cyclic. Prove that $MLST$ is cyclic if and only if $K$ is the midpoint of $BC$.

Let $O$ be the center of the equilateral triangle $ABC$. Pick two points $P_1$ and $P_2$ other than $B$, $O$, $C$ on the circle $\odot(BOC)$ so that on this circle $B$, $P_1$, $P_2$, $O$, $C$ are placed in this order. Extensions of $BP_1$ and $CP_1$ intersects respectively with side $CA$ and $AB$ at points $R$ and $S$. Line $AP_1$ and $RS$ intersects at point $Q_1$. Analogously point $Q_2$ is defined. Let $\odot(OP_1Q_1)$ and $\odot(OP_2Q_2)$ meet again at point $U$ other than $O$. Prove that $2\,\angle Q_2UQ_1 + \angle Q_2OQ_1 = 360^\circ$. Remark. $\odot(XYZ)$ denotes the circumcircle of triangle $XYZ$.

The number of positive integers $k$ for which the equation $kx - 12 = 3k$ has an integer solution for $x$ is $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7 $

Exactly one integer is written in each square of an $n$ by $n$ grid, $n \ge 3$. The sum of all of the numbers in any $2 \times 2$ square is even and the sum of all the numbers in any $3 \times 3$ square is even. Find all $n$ for which the sum of all the numbers in the grid is necessarily even.

Let $H$ be a convex, equilateral heptagon whose angles measure (in degrees) $168^\circ$, $108^\circ$, $108^\circ$, $168^\circ$, $x^\circ$, $y^\circ$, and $z^\circ$ in clockwise order. Compute the number $y$.

Vishal starts with $n$ copies of the number $1$ written on the board. Every minute, he takes two numbers $a, b$ and replaces them with either $a+b$ or $\min(a^2, b^2)$. After $n-1$ there is $1$ number on the board. Let the maximal possible value of this number be $f(n)$. Prove $2^{n/3}<f(n)\leq 3^{n/3}$.

Let \( A \) be a \( 2 \times 2 \) matrix with integer entries and \(\det A \neq 0\). If the sequence \(\operatorname{tr}(A^n)\), for \( n = 1, 2, 3, \ldots \), is bounded, show that \[ A^{12} = I \quad \text{or} \quad (A^2 - I)^2 = O. \] Here, \( I \) and \( O \) denote the identity and zero matrices, respectively, and \(\operatorname{tr}\) denotes the trace of the matrix (the sum of the elements on the main diagonal).

Let $a=256$. Find the unique real number $x>a^2$ such that \[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\] [i]Proposed by James Lin.[/i]

$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that $$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$ [i]Proposed by Ali Partofard[/i]

The best part of grandma’s $18$ cm $\times 36$ cm rectangle-shaped cake is the chocolate covering on the edges. Her three grandchildren would like to split the cake between each other so that everyone gets the same amount (of the area) of the cake, and they all get the same amount of the delicious perimeter too. a) Can they cut the cake into three convex pieces like that? b) The next time grandma baked this cake, the whole family wanted to try it so they had to cut the cake into six convex pieces this way. Is this possible? c) Soon the entire neighbourhood has heard of the delicious cake. Can the cake be cut into $12$ convex pieces with the same conditions?

Let $ G$ be a connected graph with $ n$ vertices and $ m$ edges such that each edge is contained in at least one triangle. Find the minimum value of $ m$.

Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers. (Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)

On the plane are given $ k\plus{}n$ distinct lines , where $ k>1$ is integer and $ n$ is integer as well.Any three of these lines do not pass through the same point . Among these lines exactly $ k$ are parallel and all the other $ n$ lines intersect each other.All $ k\plus{}n$ lines define on the plane a partition of triangular , polygonic or not bounded regions. Two regions are colled different, if the have not common points or if they have common points only on their boundary.A regions is called ''good'' if it contained in a zone between two parallel lines . If in a such given configuration the minimum number of ''good'' regionrs is $ 176$ and the maximum number of these regions is $ 221$, find $ k$ and $ n$. Babis