Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Find all pairs of prime numbers $p, q$ such that the number $pq + p - 6$ is also prime.

Let $ a,b$ be $ 2$ distinct positive interger number such that $ (a^2\plus{}ab\plus{}b^2)|ab(a\plus{}b)$. Prove that: $ |a\minus{}b|>\sqrt [3] {ab}$.

Let $\alpha$ and $\beta$ be irrational numbers with the property that $$\frac{1}{\alpha} +\frac{1}{\beta}=1$$ Let$\{a_n\}$ and $\{b_n\}$ be the sequences given by $a_n= \lfloor n\alpha \rfloor$ and $b_n= \lfloor n\beta \rfloor$ respectively. Prove that the sequences $\{ a_n\}$ and $\{ b_n \} $ has no term in common and cover all the natural numbers. I know this theorem from long ago, but forgot the proof of it. Can anybody help me with this?

[b]p1.[/b] Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccccc} & & & a & b & c & d \\ x & & & & & a & b \\ \hline & & c & d & b & d & b \\ + & c & e & b & f & b & \\ \hline & c & g & a & e & g & b \\ \end{tabular}$ [b]p2.[/b] $5$ numbers are placed on the circle. It is known that the sum of any two neighboring numbers is not divisible by $3$ and the sum of any three consecutive numbers is not divisible by $3$. How many numbers on the circle are divisible by $3$? [b]p3.[/b] $n$ teams played in a volleyball tournament. Each team played precisely one game with all other teams. If $x_j$ is the number of victories and $y_j$ is the number of losses of the $j$th team, show that $$\sum^n_{j=1}x^2_j=\sum^n_{j=1} y^2_j $$ [b]p4.[/b] Three cars participated in the car race: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p5.[/b] The side of the square is $4$ cm. Find the sum of the areas of the six half-disks shown on the picture. [img]https://cdn.artofproblemsolving.com/attachments/c/b/73be41b9435973d1c53a20ad2eb436b1384d69.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A$ be a set of positive integers such that for any two distinct elements $x, y\in A$ we have $|x-y| \geq \frac{xy}{25}.$ Prove that $A$ contains at most nine elements. Give an example of such a set of nine elements.

(i): Prove that if $n$ is a positive integer, then $$\binom{2n}{n}=\frac{(2n)!}{(n!)^2}$$ is a positive integer that is divisible by all prime numbers $p$ with $n<p\le 2n$, and that $$\binom{2n}{n}<2^{2n}.$$ (ii): For $x$ a positive real number, let $\pi(x)$ denote the number of prime numbers $p \le x$. [Thus, $\pi(10) = 4$ since there are $4$ primes, viz., $2$, $3$, $5$, and $7$, not exceeding $10$.]Prove that if $n \ge 3$ is an integer, then (a)$$\pi(2n) < \pi(n) + {{2n}\over{\log_2(n)}};$$(b)$$\pi(2^n) < {{2^{n+1}\log_2(n-1)}\over{n}};$$(c) Deduce that, for all real numbers $x \ge 8$,$$\pi(x) < {{4x \log_2(\log_2(x))}\over{\log_2(x)}}.$$

Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will exactly two friends visit her? $\textbf{(A) }48\qquad \textbf{(B) }54\qquad \textbf{(C) }60\qquad \textbf{(D) }66\qquad \textbf{(E) }72\qquad$

In a bank, only the manager knows the safe's combination, which is a five-digit number. To support this combination, each of the bank's ten employees is given a five-digit number. Each of these backup numbers has in one of the five positions the same digit as the combination and in the other four positions a different digit than the one in that position in the combination. Backup numbers are: $07344$, $14098$, $27356$, $36429$, $45374$, $52207$, $63822$, $70558$, $85237$, $97665$. What is the combination to the safe?

Let $x$ and $y$ be positive integers and assume that $z=\frac{4xy}{x+y}$ is an odd integer. Prove that at least one divisor of $z$ can be expressed in the form $4n-1$ where $n$ is a positive integer.

Let $a, b, c$ be positive integers such that no number divides some other number. If $ab-b+1 \mid abc+1$, prove that $c \geq b$.

[u]Round 5[/u] [b]p13.[/b] Two concentric circles have radii $1$ and $3$. Compute the length of the longest straight line segment that can be drawn froma point on the circle of radius $1$ to a point on the circle of radius $3$ if the segment cannot intersect the circle of radius $1$. [b]p14.[/b] Find the value of $\frac{1}{3} + \frac29+\frac{3}{27}+\frac{4}{81}+\frac{5}{243}+...$ [b]p15.[/b] Bob is trying to type the word "welp". However, he has a $18$ chance ofmistyping each letter and instead typing one of four adjacent keys, each with equal probability. Find the probability that he types "qelp" instead of "welp". [u]Round 6[/u] [b]p16.[/b] How many ways are there to tile a $2\times 12$ board using an unlimited supply of $1\times 1$ and $1\times 3$ pieces? [b]p17.[/b] Jeffrey and Yiming independently choose a number between $0$ and $1$ uniformly at random. What is the probability that their two numbers can formthe sidelengths of a triangle with longest side of length $1$? [b]p18.[/b] On $\vartriangle ABC$ with $AB = 12$ and $AC = 16$, let $M$ be the midpoint of $BC$ and $E$,$F$ be the points such that $E$ is on $AB$, $F$ is on $AC$, and $AE = 2AF$. Let $G$ be the intersection of $EF$ and $AM$. Compute $\frac{EG}{GF}$ . [u]Round 7[/u] [b]p19.[/b] Find the remainder when $2019x^{2019} -2018x^{2018}+ 2017x^{2017}-...+x$ is divided by $x +1$. [b]p20.[/b] Parallelogram $ABCD$ has $AB = 5$, $BC = 3$, and $\angle ABC = 45^o$. A line through C intersects $AB$ at $M$ and $AD$ at $N$ such that $\vartriangle BCM$ is isosceles. Determine the maximum possible area of $\vartriangle MAN$. [b]p21[/b]. Determine the number of convex hexagons whose sides only lie along the grid shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/9/93cf897a321dfda282a14e8f1c78d32fafb58d.png[/img] [u]Round 8[/u] [b]p22.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 4$, $BC = 5$, and $C A = 6$. Extend ray $\overrightarrow{AB}$ to a point $D$ such that $AD = 12$, and similarly extend ray $\overrightarrow{CB}$ to point $E$ such that $CE = 15$. Let $M$ and $N$ be points on the circumcircles of $ABC$ and $DBE$, respectively, such that line $MN$ is tangent to both circles. Determine the length of $MN$. [b]p23.[/b] A volcano will erupt with probability $\frac{1}{20-n}$ if it has not erupted in the last $n$ years. Determine the expected number of years between consecutive eruptions. [b]p24.[/b] If $x$ and $y$ are integers such that $x+ y = 9$ and $3x^2+4x y = 128$, find $x$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165997p28809441]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166099p28810427]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets. For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$. [i]Proposed by Victor Wang[/i]

The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = 0$, $a_n = a_{[n/2]} + (-1)^{n(n+1)/2}$. Show that for any positive integer $k$ we can find $n$ in the range $2^k \leq n < 2^{k+1}$ such that $a_n = 0$.

Let $ k$ be a positive integer. Prove that the number $ (4 \cdot k^2 \minus{} 1)^2$ has a positive divisor of the form $ 8kn \minus{} 1$ if and only if $ k$ is even. [url=http://www.mathlinks.ro/viewtopic.php?p=894656#894656]Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above.[/url] [i]Author: Kevin Buzzard and Edward Crane, United Kingdom [/i]

Show that for every prime number $p$ and every positive integer $n\geq2$ there exists a positive integer $k$ such that the decimal representation of $p^k$ contains $n$ consecutive equal digits.

Prove that if $ n $ is a non-negative integer, then number $$ 2^{n+2} + 3^{2n+1}$$ is divisible by $7$.

Let $n$ be a positive integer. a) Prove that the number $(2n + 1)^3 - (2n - 1)^3$ is the sum of three perfect squares. b) Prove that the number $(2n+1)^3-2$ is the sum of $3n-1$ perfect squares greater than $1$.

Let $ S$ be a set of all points of a plane whose coordinates are integers. Find the smallest positive integer $ k$ for which there exists a 60-element subset of set $ S$ with the following condition satisfied for any two elements $ A,B$ of the subset there exists a point $ C$ contained in $ S$ such that the area of triangle $ ABC$ is equal to k .

Find all integer triples $(p,m,n)$ that satisfy: $p^m-n^3=27$ where $p$ is a prime number.

Find all function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ , $(f(a)+b) f(a+f(b))=(a+f(b))^2$

In each unit cell of a finite set of cells of an infinite checkered board, an integer is written so that the sum of the numbers in each row, as well as in each column, is divided by $2002$. Prove that every number $\alpha$ can be replaced by a certain number $\alpha'$ , divisible by $2002$ so that $|\alpha-\alpha'| <2002$ and the sum of the numbers in all rows, and in all columns will not change.

A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

The pairwise greatest common divisors of five positive integers are $$2, 3, 4, 5, 6, 7, 8, p, q, r$$ in some order, for some positive integers $p, q, r$. Compute the minimum possible value of $p + q + r$.

Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that: \[a^p+b^p=p^c.\]