Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$

Let $n \geq 2$ be a positive integer. Show that there are exactly $2^{n-3}n(n-1)$ $n$-tuples of integers $(a_1,a_2,\dots,a_n)$, which satisfy the conditions: (i) $a_1=0$; (ii) for each $m$, $2 \leq m \leq n$, there is an index in $m$, $1 \leq i_m <m$, such that $\left|a_{i_m}-a_m\right|\leq 1$; (iii) the $n$-tuple $(a_1,a_2,\dots,a_n)$ contains exactly $n-1$ different numbers.

We are having  99 equal circles in a row and in the interior, we write inside them all the numbers from 1 up to 99 (one number in each circle).We color each of the circles with one of the two colors available: red and green. A coloring is good if it has the ability: Red circles lying in the interval of the numbers from 1 up to 50  are more than the red circles lying in the interval of the numbers from 51  up to 99 . a) Find how many different colorings can be constructed. b) Find how many different good colorings can be constructed. (Note: Two colorings are different, if they have different color in at least one of their circles.)

Mark has six boxes lined up in a straight line. Inside each of the first three boxes are a red ball, a blue ball, and a green ball. He randomly selects a ball from each of the three boxes and puts them into a fourth box. Then, he randomly selects a ball from each of the four boxes and puts them into a fifth box. Next, he randomly selects a ball from each of the five boxes and puts them into a sixth box, arriving at three boxes with $1, 3,$ and $5$ balls, respectively. The probability that the box with $3$ balls has each type of color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by [b]AOPS12142015[/b][/i]

How many integers can be expressed in the form: $\pm 1 \pm 2 \pm 3 \pm 4 \pm \cdots \pm 2018$?

$25$ students are on exams. Exam consists of some questions with $5$ variants of answer. Every two students gave same answer for not more than $1$ question. Prove, that there are not more than $6$ questions in exam.

If integers $a$, $b$, $c$, and $d$ satisfy $ bc + ad = ac + 2bd = 1 $, find all possible values of $ \frac {a^2 + c^2}{b^2 + d^2} $.

Let the four real solutions to the equation $x^2 + \frac{144}{x^2} = 25$ be $r_1, r_2, r_3$, and $r_4$. Find $|r_1| +|r_2| +|r_3| +|r_4|$.

A circle cuts off four right-angled triangles from rectangle $ABCD$.Let $A_0, B_0, C_0$ and $D_0$ be the midpoints of the correspondent hypotenuses. Prove that $A_0C_0 = B_0D_0$ [i]Proosed by L.Shteingarts[/i]

For how many ordered pairs $(x,y)$ of integers is it true that $0<x<y<10^{6}$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y?$

Let $k>1$ be an integer, $p=6k+1$ be a prime number and $m=2^{p}-1$ . Prove that $\frac{2^{m-1}-1}{127m}$ is an integer.

Triangle $ABC$ has circumcircle $\Omega$ and incircle $\omega$, where $\omega$ is tangent to $BC, CA, AB$ at $D,E,F$, respectively. $T$ is an arbitrary point on $\omega$. $EF$ meets $BC$ at $K$, $AT$ meets $\Omega$ again at $P$, $PK$ meets $\Omega$ again at $S$. $X$ is a point on $\Omega$ such that $S, D, X$ are colinear. Let $Y$ be the intersection of $AX$ and $EF$, prove that $YT$ is tangent to $\omega$. [i]Proposed by chengbilly[/i]

Mikko takes a multiple choice test with ten questions. His only goal is to pass the test, and this requires seven points. A correct answer is worth one point, and answering wrong results in the deduction of one point. Mikko knows for sure that he knows the correct answer in the six first questions. For the rest, he estimates that he can give the correct answer to each problem with probability $p, 0 < p < 1$. How many questions Mikko should try?

Let $A_1, \ldots, A_k$ be $n\times n$ idempotent complex matrices such that $A_iA_j = -A_iA_j$ for all $1 \leq i < j \leq k$. Prove that at least one of the matrices has rank not exceeding $\frac{n}{k}$.

Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?

Let $n \ge 2$ be a fixed integer. The cells of an $n \times n$ table are filled with the integers from $1$ to $n^2$ with each number appearing exactly once. Let $N$ be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite vertices of this rectangle. Find the largest possible value of $N$. [i]Anonymous[/i]

If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$.

In a parallelogram $ABCD$, the perpendiculars from $A$ to $BC$ and $CD$ meet the line segments $BC$ and $CD$ at the points $E$ and $F$ respectively. Suppose $AC = 37$ cm and $EF = 35$ cm. Let $H$ be the orthocentre of $\vartriangle AEF$. Find the length of $AH$ in cm. Show the steps in your calculations.

Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom there is exactly enough cheese if each customer will buy a portion of cheese of weight exactly equal to the average weight of the previous purchases. Could it happen that the salesgirl can declare, after each of the first $10$ customers has made their purchase, that there just enough cheese for the next $10$ customers? If so, how much cheese will be left in the store after the first $10$ customers have made their purchases? (The average weight of a series of purchases is the total weight of the cheese sold divided by the number of purchases.)

Let $ABC$ be an acute-angled triangle with altitudes $AD,BE,$ and $CF$. Let $H$ be the orthocentre, that is, the point where the altitudes meet. Prove that \[\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.\]

Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$. [i]Proposed by Fernando Campos, Mexico[/i]

Zeroes are written in all cells of a $5 \times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. Is it possible to obtain the number 2012 in all cells simultaneously?

İn triangle $ABC$ the bisector of $\angle BAC$ intersects the side $BC$ at the point $D$.The circle $\omega $ passes through $A$ and tangent to the side $BC$ at $D$.$AC$ and $\omega $ intersects at $M$ second time , $BM$ and $\omega $ intersects at $P$ second time. Prove that point $P$ lies on median of triangle $ABD$.

At least three players have participated in a tennis tournament. Evey two players have played each other exactly once, and each player has at least one match won. Show that there are three players $A,B,C$ such that $A$ won against $B$, $B$ won against $C$ and $C$ won against $A$.

Find the greatest possible number of lines in space that all pass through a single point and the angle between any two of them is the same.