Find all quadruples of positive integers \( (a, p, q, r) \), where \( p, q, r \) are prime numbers, such that the following equation holds: \[ p^2q^2 + q^2r^2 + r^2p^2 + 3 = 4 \cdot 13^a. \] [i]Proposed by Oleksii Masalitin[/i]

Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened. Decide whether there exists a strategy for player $A$ to win in a finite number of moves.

Given any odd integer $n>3$ that is not divisible by $3$, determine whether it is possible to fill an $n \times n$ grid with $n^2$ integers such that (each cell filled with a number, the number at the intersection of the $i$-th row and $j$-th column is denoted as $a_{ij}$): $\cdot$ Each row and each column contains a permutation of the numbers $0,1,2, \cdots, n-1$. $\cdot$ The pairs $(a_{ij},a_{ji})$ for $i<j$ are all distinct.

Suppose that $x,y$ are distinct positive reals, and $n>1$ is a positive integer. If \[x^n-y^n=x^{n+1}-y^{n+1},\] then show that \[1<x+y<\frac{2n}{n+1}.\]

Let $ABC$ and $AMN$ be two similar, non-overlapping triangles with the same orientation, such that $AB=AC$ and $AM=AN$. Let $O$ be the circumcentre of the triangle $MAB$. Prove that the points $O, C, N$ and $A$ lie on a circle if and only if the triangle $ABC$ is equilateral.

Point $M$ is middle of side $AB$ of equilateral triangle $ABC$. Points $D$ and $E$ lie on segments $AC$ and $BC$, respectively and $\angle DME = 60 ^{\circ}$. Prove that, $AD + BE = DE + \frac{1}{2}AB$.

Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.

Let $ABC$ be a triangle and $D$ be a point on segment $AC$. The circumscribed circle of the triangle $BDC$ cuts $AB$ again at $E$ and the circumference circle of the triangle $ABD$ cuts $BC$ again at $F$. Prove that $AE = CF$ if and only if $BD$ is the interior bisector of $\angle ABC$.

Let $k$ be a given integer, $3 < k \leq n$. Consider a graph $G$ with $n$ vertices satisfying the condition: for any two non-adjacent vertices $x$ and $y$ in graph $G$, the sum of their degrees must satisfy $d(x) + d(y) \geq k$. Please answer the following questions and prove your conclusions. (1) Suppose the length of the longest path in graph $G$ is $l$ satisfying the inequality $3 \leq l < k$, does graph $G$ necessarily contain a cycle of length $l+1$? (The length of a path or cycle refers to the number of edges that make up the path or cycle.) (2) For the case where $3 < k \leq n-1$ and graph $G$ is connected, can we determine that the length of the longest path in graph $G$, $l \geq k$? (3) For the case where $3 < k = n-1$, is it necessary for graph $G$ to have a path of length $n-1$ (i.e., a Hamiltonian path)?

Be $R$ a positive real number. If $R, 1, R+\frac12$ are triangle sides, call $\theta$ the angle between $R$ and $R+\frac12$ (in rad). Prove $2R\theta$ is between $1$ and $\pi$.

$10$ people attended a party. For every $3$ people, there exist at least $2$ people who don’t know each other. Prove that there exist $4$ people who don’t know each other.

An $n\times n$ square ($n$ is a positive integer) consists of $n^2$ unit squares.A $\emph{monotonous path}$ in this square is a path of length $2n$ beginning in the left lower corner of the square,ending in its right upper corner and going along the sides of unit squares. For each $k$, $0\leq k\leq 2n-1$, let $S_k$ be the set of all the monotonous paths such that the number of unit squares lying below the path leaves remainder $k$ upon division by $2n-1$.Prove that all $S_k$ contain equal number of elements.

For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.

Do there exist two positive reals $\alpha, \beta$ such that each positive integer appears exactly once in the following sequence? \[ 2020, [\alpha], [\beta], 4040, [2\alpha], [2\beta], 6060, [3\alpha], [3\beta], \cdots \] If so, determine all such pairs; if not, prove that it is impossible.

Suppose \[ n(n\plus{}1)a_{n\plus{}1}\equal{}n(n\minus{}1)a_n\minus{}(n\minus{}2)a_{n\minus{}1} \] for every positive integer $ n\ge1$. Given that $ a_0\equal{}1,a_1\equal{}2$, find \[ \frac{a_0}{a_1}\plus{}\frac{a_1}{a_2}\plus{}\frac{a_2}{a_3}\plus{}\dots\plus{}\frac{a_{50}}{a_{51}}. \]

Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade? $ \textbf{(A) } 129 \qquad \textbf{(B) } 137 \qquad \textbf{(C) } 174 \qquad \textbf{(D) } 223 \qquad \textbf{(E) } 411$