Find the natural number $ n $ knowing that the sum $$ 1 + 2 + 3 + \ldots + n$$ is a three-digit number with identical digits.

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ such that for all prime $p$ the following condition holds: $$p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a)$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

Given a prime number $p$, a set is said to be $p$-good if the set contains exactly three elements $a, b, c$ and $a + b \equiv c \pmod{p}$. Find all prime number $p$ such that $\{ 1, 2, \cdots, p-1 \}$ can be partitioned into several $p$-good sets. [i]Proposed by capoouo[/i]

Let $b$ and $c$ be any two positive integers. Define an integer sequence $a_n$, for $n\geq 1$, by $a_1=1$, $a_2=1$, $a_3=b$ and $a_{n+3}=ba_{n+2}a_{n+1}+ca_n$. Find all positive integers $r$ for which there exists a positive integer $n$ such that the number $a_n$ is divisible by $r$.

Determine all sequences $a_1, a_2, a_3, \dots$ of nonnegative integers such that $a_1 < a_2 < a_3 < \dots$ and $a_n$ divides $a_{n - 1} + n$ for all $n \geq 2$.

Compute gcd$(5^{2547} - 1, 5^{2004} - 1)$.

For each positive integer $ k$ and $ n,$ let $ S_k(n)$ be the base $ k$ digit sum of $ n.$ Prove that there are at most two primes $ p$ less than $20,000$ for which $ S_{31}(p)$ are composite numbers with at least two distinct prime divisors.

Let $ n$ be a given positive integer. Show that we can choose numbers $ c_k\in\{\minus{}1,1\}$ ($ i\le k\le n$) such that \[ 0\le\sum_{k\equal{}1}^nc_k\cdot k^2\le4.\]

Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [i]Author: Ray Li[/i]

Find all natural number which can be expressed in $ \frac{a\plus{}b}{c}\plus{}\frac{b\plus{}c}{a}\plus{}\frac{c\plus{}a}{b}$ where $ a,b,c\in \mathbb{N}$ satisfy $ \gcd(a,b)\equal{}\gcd(b,c)\equal{}\gcd(c,a)\equal{}1$

Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.

In this problem, we consider only polynomials with integer coeffients. Call two polynomials $p$ and $q$ [i]really close[/i] if $p(2k + 1) \equiv q(2k + 1)$ (mod $210$) for all $k \in Z^+$. Call a polynomial $p$ [i]partial credit[/i] if no polynomial of lesser degree is [i]really close[/i] to it. What is the maximum possible degree of [i]partial credit[/i]?

$n \geq k$ -two natural numbers. $S$ -such natural, that have not less than $n$ divisors. All divisors of $S$ are written in descending order. What minimal number of divisors can have number from $k$-th place ?

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

If $m, n$, and $p$ are three different natural numbers, each between $2$ and $9$, what then are all the possible integer value(s) of the expression $\frac{m+n+p}{m+n}$?

[b]p1.[/b] An evil galactic empire is attacking the planet Naboo with numerous automatic drones. The fleet defending the planet consists of $101$ ships. By the decision of the commander of the fleet, some of these ships will be used as destroyers equipped with one rocket each or as rocket carriers that will supply destroyers with rockets. Destroyers can shoot rockets so that every rocket destroys one drone. During the attack each carrier will have enough time to provide each destroyer with one rocket but not more. How many destroyers and how many carriers should the commander assign to destroy the maximal number of drones and what is the maximal number of drones that the fleet can destroy? [b]p2.[/b] Solve the inequality: $\sqrt{x^2-3x+2} \le \sqrt{x+7}$ [b]p3.[/b] Find all positive real numbers $x$ and $y$ that satisfy the following system of equations: $$x^y = y^{x-y}$$ $$x^x = y^{12y}$$ [b]p4.[/b] A convex quadrilateral $ABCD$ with sides $AB = 2$, $BC = 8$, $CD = 6$, and $DA = 7$ is divided by a diagonal $AC$ into two triangles. A circle is inscribed in each of the obtained two triangles. These circles touch the diagonal at points $E$ and $F$. Find the distance between the points $E$ and $F$. [b]p5.[/b] Find all positive integer solutions $n$ and $k$ of the following equation: $$\underbrace{11... 1}_{n} \underbrace{00... 0}_{2n+3} + \underbrace{77...7}_{n+1} \underbrace{00...0}_{n+1}+\underbrace{11...1}_{n+2} = 3k^3.$$ [b]p6.[/b] The Royal Council of the planet Naboo consists of $12$ members. Some of these members mutually dislike each other. However, each member of the Council dislikes less than half of the members. The Council holds meetings around the round table. Queen Amidala knows about the relationship between the members so she tries to arrange their seats so that the members that dislike each other are not seated next to each other. But she does not know whether it is possible. Can you help the Queen in arranging the seats? Justify your answer. PS. You should use hide for answers.

Is it possible to color points in the Cartesian Plane $(x,y)$ with integer coordinates with three colors, such that each color appears infinitely many times on infinitely many lines parallel to the $x$-axis and that any three points, each of a different color, are not in a line? Justify your answer.

On a blackboard the numbers $1,2,3,\dots,170$ are written. You want to color each of these numbers with $k$ colors $C_1,C_2, \dots, C_k$, such that the following condition is satisfied: for each $i$ with $1 \leq i < k$, the sum of all numbers with color $C_i$ divide the sum of all numbers with color $C_{i+1}$. Determine the largest possible value of $k$ for which it is possible to do that coloring.

Define the sequence $(a_n)_{n=1}^\infty$ of positive integers by $a_1=1$ and the condition that $a_{n+1}$ is the least integer such that \[\mathrm{lcm}(a_1, a_2, \ldots, a_{n+1})>\mathrm{lcm}(a_1, a_2, \ldots, a_n)\mbox{.}\] Determine the set of elements of $(a_n)$.

Let $N$ be the number of integral solutions of the equation \[x^2 - y^2 = z^3 - t^3\] satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation \[x^2 - y^2 = z^3 - t^3 + 1\] satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$

a) Given a quartet of positive numbers $(a,b,c,d)$. It is transformed to the new one according to the rule: $a'=ab, b' =bc, c'=cd,d'=da$. The second one is transformed to the third according to the same rule and so on. Prove that if at least one initial number does not equal 1, than You can never obtain the initial set. b) Given a set of $2^k$ ($k$-th power of two) numbers, equal either to $1$ or to $-1$. It is transformed as that was in the a) problem (each one is multiplied by the next, and the last -- by the first. Prove that You will always finally obtain the set of positive units.

Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.

Let $P$ be a finite set of primes, $A$ an infinite set of positive integers, where every element of $A$ has a prime factor not in $P$. Prove that there exist an infinite subset $B$ of $A$, such that the sum of elements in any finite subset of $B$ has a prime factor not in $P$.

Prove that in the decimal notation of the number $(5+\sqrt{26})^{-1973}$ immediately after the decimal point there are at least $1973$ zeros.