Given a regular polygon with $p$ sides, where $p$ is a prime number. After rotating this polygon about its center by an integer number of degrees it coincides with itself. What is the maximal possible number for $p$?

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

Are there $10$ different integers such that all the sums made up of $9$ of them are perfect squares?

Consider the sequence of integers $ \left( a_n\right)_{n\ge 0} $ defined as $$ a_n=\left\{\begin{matrix}n^6-2017, & 7|n\\ \frac{1}{7}\left( n^6-2017\right) , & 7\not | n\end{matrix}\right. . $$ Determine the largest length a string of consecutive terms from this sequence sharing a common divisor greater than $ 1 $ may have.

[u]Set 8[/u] [b]p22.[/b] For monic quadratic polynomials $P = x^2 + ax + b$ and $Q = x^2 + cx + d$, where $1 \le a, b, c, d \le 10$ are integers, we say that $P$ and $Q$ are friends if there exists an integer $1 \le n \le 10$ such that $P(n) = Q(n)$. Find the total number of ordered pairs $(P, Q)$ of such quadratic polynomials that are friends. [b]p23.[/b] A three-dimensional solid has six vertices and eight faces. Two of these faces are parallel equilateral triangles with side length $1$, $\vartriangle A_1A_2A_3$ and $\vartriangle B_1B_2B_3$. The other six faces are isosceles right triangles — $\vartriangle A_1B_2A_3$, $\vartriangle A_2B_3A_1$, $\vartriangle A_3B_1A_2$, $\vartriangle B_1A_2B_3$, $\vartriangle B_2A_3B_1$, $\vartriangle B_3A_1B_2$ — each with a right angle at the second vertex listed (so for instace $\vartriangle A_1B_2A_3$ has a right angle at $B_2$). Find the volume of this solid. [b]p24.[/b] The digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ are each colored red, blue, or green. Find the number of colorings such that any integer $ n \ge 2$ has that (a) If $n$ is prime, then at least one digit of $n$ is not blue. (b) If $n$ is composite, then at least one digit of $n$ is not green. PS. You should use hide for answers.

For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of several distinct positive integers not exceeding $2n$?

let m and n be natural numbers such that: $3m|(m+3)^n+1$ Prove that $\frac{(m+3)^n+1}{3m}$ is odd

Find the remainder when $2^{2019}$ is divided by $7$.

Prove that if $x,y$ and $z$ are pairwise relatively prime positive integers, and if $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$, then $x+y, x-z, y-z$ are perfect squares of integers.

Let $p$ be a prime number such that $p\equiv 1\pmod 9$. Show that there exist an integer $n$ such that $n^3-3n+1$ is divisible by $p$.

Suppose that $\alpha$ is a real number and $a_1<a_2<.....$ is a strictly increasing sequence of natural numbers such that for each natural number $n$ we have $a_n\le n^{\alpha}$. We call the prime number $q$ golden if there exists a natural number $m$ such that $q|a_m$. Suppose that $q_1<q_2<q_3<.....$ are all the golden prime numbers of the sequence $\{a_n\}$. [b]a)[/b] Prove that if $\alpha=1.5$, then $q_n\le 1390^n$. Can you find a better bound for $q_n$? [b]b)[/b] Prove that if $\alpha=2.4$, then $q_n\le 1390^{2n}$. Can you find a better bound for $q_n$? [i]part [b]a[/b] proposed by mahyar sefidgaran by an idea of this question that the $n$th prime number is less than $2^{2n-2}$ part [b]b[/b] proposed by mostafa einollah zade[/i]

Find all collections of $63$ integer numbers such that the square of each number is equal to the sum of all other numbers, and not all the numbers are equal. (O. Rudenko)

Let $N\ge2$ be a positive integer. Given a sequence of natural numbers $a_1,a_2,a_3,\dots,a_{N+1}$ such that for every integer $1\le i\le j\le N+1$, $$a_ia_{i+1}a_{i+2}\dots a_j \not\equiv1\mod{N}$$ Prove that there exist a positive integer $k\le N+1$ such that $\gcd(a_k, N) \neq 1$

Let $m$ be the number of ordered solutions $(a,b,c,d,e)$ satisfying: $1)$ $a,b,c,d,e\in \mathbb{Z}^{+}$; $2)$ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}=1$; Prove that $m$ is odd.

$a$ is an integer and $p$ is a prime number and we have $p\ge 17$. Suppose that $S=\{1,2,....,p-1\}$ and $T=\{y|1\le y\le p-1,ord_p(y)<p-1\}$. Prove that there are at least $4(p-3)(p-1)^{p-4}$ functions $f:S\longrightarrow S$ satisfying $\sum_{x\in T} x^{f(x)}\equiv a$ $(mod$ $p)$. [i]proposed by Mahyar Sefidgaran[/i]

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

Find all natural numbers $n$ such that $\frac{\sqrt{n}}{2}+\frac{10}{\sqrt{n}}$ is a natural number.

Nir finds integers $a_0, a_1, ... , a_{208}$ such that $$(x + 2)^{208} = a_0x^0 + a_1x^1 + a_2x^2 +... + a_{208}x^{208}.$$ Let $S$ be the sum of all an such that $n -3$ is divisible by $5$. Compute the remainder when $S$ is divided by $103$.

[b]p1.[/b] Let $A\% B = BA - B - A + 1$. How many digits are in the number $1\%(3\%(3\%7))$ ? [b]p2. [/b]Three circles, of radii $1, 2$, and $3$ are all externally tangent to each other. A fourth circle is drawn which passes through the centers of those three circles. What is the radius of this larger circle? [b]p3.[/b] Express $\frac13$ in base $2$ as a binary number. (Which, similar to how demical numbers have a decimal point, has a “binary point”.) [b]p4. [/b] Isosceles trapezoid $ABCD$ with $AB$ parallel to $CD$ is constructed such that $DB = DC$. If $AD = 20$, $AB = 14$, and $P$ is the point on $AD$ such that $BP + CP$ is minimized, what is $AP/DP$? [b]p5.[/b] Let $f(x) = \frac{5x-6}{x-2}$ . Define an infinite sequence of numbers $a_0, a_1, a_2,....$ such that $a_{i+1} = f(a_i)$ and $a_i$ is always an integer. What are all the possible values for $a_{2014}$ ? [b]p6.[/b] $MATH$ and $TEAM$ are two parallelograms. If the lengths of $MH$ and $AE$ are $13$ and $15$, and distance from $AM$ to $T$ is $12$, find the perimeter of $AMHE$. [b]p7.[/b] How many integers less than $1000$ are there such that $n^n + n$ is divisible by $5$ ? [b]p8.[/b] $10$ coins with probabilities of $1, 1/2, 1/3 ,..., 1/10$ of coming up heads are flipped. What is the probability that an odd number of them come up heads? [b]p9.[/b] An infinite number of coins with probabilities of $1/4, 1/9, 1/16, ...$ of coming up heads are all flipped. What is the probability that exactly $ 1$ of them comes up heads? [b]p10.[/b] Quadrilateral $ABCD$ has side lengths $AB = 10$, $BC = 11$, and $CD = 13$. Circles $O_1$ and $O_2$ are inscribed in triangles $ABD$ and $BDC$. If they are both tangent to $BD$ at the same point $E$, what is the length of $DA$ ? PS. You had better use hide for answers.

Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal.

If $n$ is composite, prove that $\phi(n) \le n- \sqrt{n}$.

Is it possible for any finite group of people to choose a positive integer $N$ and assign a positive integer to each person in the group such that the product of two persons' number is divisible by $N$ if and only if they are friends?

It is known that some \(d\) distinct divisors of a positive integer number \(n\) form an arithmetic progression. Prove that the number \(n\) has at least \(2d - 2\) divisors. [i]Proposed by Anton Trygub[/i]

Let $n \in \mathbb{N}$ be such that $gcd(n, 6) = 1$. Let $a_1 < a_2 < \cdots < a_n$ and $b_1 < b_2 < \cdots < b_n$ be two collection of positive integers such that $a_j + a_k + a_l = b_j + b_k + b_l$ for all integers $1 \le j < k < l \le n$. Prove that $a_j = b_j$ for all $1 \le j \le n$.

Let $ S$ be the set of all positive integers whose all digits are $ 1$ or $ 2$. Denote $ T_{n}$ as the set of all integers which is divisible by $ n$, then find all positive integers $ n$ such that $ S\cap T_{n}$ is an infinite set.