A non-empty subset of $\{1,2, ..., n\}$ is called [i]arabic [/i] if arithmetic mean of its elements is an integer. Show that the number of arabic subsets of $\{1,2, ..., n\}$ has the same parity as $n$.

[u]Bases[/u] Many of you may be familiar with the decimal (or base $10$) system. For example, when we say $2013_{10}$, we really mean $2\cdot 10^3+0\cdot 10^2+1\cdot 10^1+3\cdot 10^0$. Similarly, there is the binary (base $2$) system. For example, $11111011101_2 = 1 \cdot 2^{10}+1 \cdot 2^9+1 \cdot 2^8+1 \cdot 2^7+1 \cdot 2^6+0 \cdot 2^5+1 \cdot 2^4+1 \cdot 2^3+1 \cdot 2^2+0 \cdot 2^1+1 \cdot 2^0 = 2013_{10}.$ In general, if we are given a string $(a_na_{n-1} ... a_0)_b$ in base $b$ (the subscript $b$ means that we are in base $b$), then it is equal to $\sum^n_{i=0} a_ib^i$. It turns out that for every positive integer $b > 1$, every positive integer $k$ has a unique base $b$ representation. That is, for every positive integer $k$, there exists a unique $n$ and digits $0 \le a_0,..., a_n < b$ such that $(a_na_{n-1} ... a_0)_b = k$. We can adapt this to bases $b < -1$. It actually turns out that if $b < -1$, every nonzero integer has a unique base b representation. That is, for every nonzero integer $k$, there exists a unique $n$ and digits $0 \le a_0,..., a_n < |b|$ such that $(a_na_{n-1} ... a_0)_b = k$. The next five problems involve base $-4$. Note: Unless otherwise stated, express your answers in base $10$. [b]p6.[/b] Evaluate $1201201_{-4}$. [b]p7.[/b] Express $-2013$ in base $-4$. [b]p8.[/b] Let $b(n)$ be the number of digits in the base $-4$ representation of $n$. Evaluate $\sum^{2013}_{i=1} b(i)$. [b]p9.[/b] Let $N$ be the largest positive integer that can be expressed as a $2013$-digit base $-4$ number. What is the remainder when $N$ is divided by $210$? [b]p10.[/b] Find the sum of all positive integers $n$ such that there exists an integer $b$ with $|b| \ne 4$ such that the base $-4$ representation of $n$ is the same as the base $b$ representation of $n$.

Let $n>2$ be a positive integer and $b=2^{2^n}$. Let $a$ be an odd positive integer such that $a\le b \le 2a$. Show that $a^2+b^2-ab$ is not a square.

Is it possible to choose five different positive integers so that the sum of any three of them is a prime number?

Let $n$ be a positive integer. Suppose that the diophantine equation \[z^n = 8 x^{2009} + 23 y^{2009} \] uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.

Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.

Prove that the equation $ (x\plus{}y)^n\equal{}x^m\plus{}y^m$ has a unique solution in integers with $ x>y>0$ and $ m,n>1$.

Let $n$ be a positive integer. Let $s: \mathbb N \to \{1, \ldots, n\}$ be a function such that $n$ divides $m-s(m)$ for all positive integers $m$. Let $a_0, a_1, a_2, \ldots$ be a sequence such that $a_0=0$ and \[a_{k}=a_{k-1}+s(k) \text{ for all }k\ge 1.\] Find all $n$ for which this sequence contains all the residues modulo $(n+1)^2$. [i]Proposed by N.V. Tejaswi[/i]

Let $a,b,c,d,e,d,e,f$ be positive integers such that \(\dfrac a b < \dfrac c d < \dfrac e f\). Suppose \(af-be=-1\). Show that \(d \geq b+f\).

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that [list=1] [*] $A\cap B=\{1\};$ [*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$; [*] each prime number is a divisor of some number in $A$ and also some number in $B$; [*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$. [*] Each set has infinitely many composite numbers. [/list]

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.

For every lattice point $(x, y)$ with $x, y$ non-negative integers, a square of side $\frac{0.9}{2^x5^y}$ with center at the point $(x, y)$ is constructed. Compute the area of the union of all these squares.

Find the minimum natural number $n$ with the following property: between any collection of $n$ distinct natural numbers in the set $\{1,2, \dots,999\}$ it is possible to choose four different $a,\ b,\ c,\ d$ such that: $a + 2b + 3c = d$.

The numbers from $ 51$ to $ 150$ are arranged in a $ 10\times 10$ array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers $ a$ and $ b$, at least one of the equations $ x^2 \minus{} ax \plus{} b \equal{} 0$ and $ x^2 \minus{} bx \plus{} a \equal{} 0$ has two integral roots?

Prove that there is a multiple of $2^{2022}$ that has $2022$ digits, and can be written using digits $1$ and $2$ only.

Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer. Prove that $M(a,b)$ is a square. (b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.

Let $F_n$ be the $n$th Fibonacci number ($F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$). Construct infinitely many positive integers $n$ such that $n$ divides $F_{F_n}$ but $n$ does not divide $F_n$.

Find all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integer numbers $n$, such that $$p^2 = q^2 + r^n$$ (Walther Janous)

Let $19$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $180$. Prove that not all numbers on the paper are different.