In the adjacent figure, can the numbers $1,2,3, 4,..., 18$ be placed, one on each line segment, such that the sum of the numbers on the three line segments meeting at each point is divisible by $3$?

Prove that there exists $N\in\mathbb{N}$ so that for all integer $n > N$, one may find $2019$ pairwise co-prime positive integers with \[n=a_1+a_2+\cdots+a_{2019}\] and \[2019<a_1<a_2<\cdots<a_{2019}\]

Let for natural numbers $a,b,c$ and any natural $n$ we have that $(abc)^n$ divides $ ((a^n-1)(b^n-1)(c^n-1)+1)^3$. Prove that then $a=b=c$.

Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.

Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.

Find the two last digits of $2012^{2012}$.

Solve equation $(3a-bc)(3b-ac)(3c-ab)=1000$ in integers. (V.Brayman)

Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have: $$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$ $$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$ Then we have : $$bP(\frac{a}{c})=dQ(\frac{a}{c})$$ (Two polynomials are relatively prime if they don't have a common root) Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]

Determine all triplets of real numbers $(x, y, z)$ that satisfy the equation $4xyz = x^4 + y^4 + z^4 + 1$.

Let $a$ and $n>0$ be integers. Define $a_{n} = 1+a+a^2...+a^{n-1}$. Show that if $p|a^p-1$ for all prime divisors of $n_{2}-n_{1}$, then the number $\frac{a_{n_{2}}-a_{n_{1}}}{n_{2}-n_{1}}$ is an integer.

Four frogs are positioned at four points on a straight line such that the distance between any two neighbouring points is 1 unit length. Suppose the every frog can jump to its corresponding point of reflection, by taking any one of the other 3 frogs as the reference point. Prove that, there is no such case that the distance between any two neighbouring points, where the frogs stay, are all equal to 2008 unit length.

A sequence of real numbers is called a [i]Fibonacci [/i] sequence if $$t_{n+2} = t_{n+1} + t_n$$ for $n= 1,2,3,. .$ . Two Fibonacci sequences are said to be [i]essentially different[/i] if the terms of one sequence cannot be obtained by multiplying the terms of the other by a constant. For example, the Fibonacci sequences $1,2,3,5,8,...$ and $1,3,4,7,11,...$ are essentially different, but the sequences $1,2,3,5,8,...$ and $2,4,6,10,16,...$ are not. (a) Prove that there exist real numbers $p$ and $q$ such that the sequences $1,p,p^2,p^3,...$ and $1,q,q^2,q^3,...$ are essentially different Fibonacci sequences. (b) Let $a_1,a_2,a_3,...$ and $b_1,b_2,b_3,...$ be essentially different Fibonacci sequences. Prove that for every Fibonacci sequence $t_1,t_2,t_3,...$, there exists exactly one number $\alpha$ and exactly one number $\beta$, such that: $$t_n = \alpha a_n + \beta b_n$$ for $n = 1,2,3,...$ (c) $t_1,t_2,t_3,...$, is the Fibonacci sequence with $t_1 = 1$ and $t_2= 2$. Express $t_n$ in terms of $n$.

Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).

Find all prime numbers $p$ and $q$ such that $$1 + \frac{p^q - q^p}{p + q}$$ is a prime number. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Given positive integer $k$, prove that there exists a positive integer $N$ depending only on $k$ such that for any integer $n\geq N$, $\binom{n}{k}$ has at least $k$ different prime divisors.

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

Nadya has $2022$ cards, each with a number one or seven written on it. It is known that there are both cards.Nadya looked at all possible $2022$-digit numbers that can be composed from all these cards. What is the largest value that can take the greatest common divisor of all these numbers?

Find integer solutions $(x,y)$ of the equation ($1964$ times "$\sqrt{}$"): $$\sqrt{x+\sqrt{x+\sqrt{....\sqrt{x+\sqrt{x}}}}}=y$$

A sequence of integers $a_0, a_1 …$ is called [i]kawaii[/i] if $a_0 =0, a_1=1,$ and $$(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0$$ for all integers $n \geq 0$. An integer is called [i]kawaii[/i] if it belongs to some kawaii sequence. Suppose that two consecutive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that $m$ is divisible by $3,$ and that $m/3$ is also kawaii.

Find all prime number $p$, such that there exist an infinite number of positive integer $n$ satisfying the following condition: $p|n^{ n+1}+(n+1)^n.$ (September 29, 2012, Hohhot)

Let $T = \{n \in N|$n consists of $2$ digits $\}$ and $$P = \{x|x = n(n + 1)... (n + 7); n,n + 1,..., n + 7 \in T\}.$$ Determine the gcd of the elements of $P$.

Find the smallest positive integer $N$ of two or more digits that has the following property: If we insert any non-null digit $d$ between any two adjacent digits of $N$ we obtain a number that is a multiple of $d$.

Suppose $p$ is a prime number and $x, y, z$ are integers satisfying $0 < x < y < z <p$. If $x^3, y^3, z^3$ have equal remainders when divided by $p$, prove that $x ^ 2 + y ^ 2 + z ^ 2$ is divisible by $x + y + z$.

The polynomial $P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9$ has the property that $P(\tfrac{1}{k})=\tfrac{1}{k}$ for $k=1,2,3,4,5,6,7,8,9$. There are relatively prime positive integers $m$ and $n$ such that $P(\tfrac{1}{10})=\tfrac{m}{n}$. Find $n-10m$.

Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there ?