Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

Solve in nonnegative integers the following equation : $$21^x+4^y=z^2$$

Define a sequence $\{a_n\}_{n\ge 1}$ such that $a_1=1,a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $\text{gcd}(m,a_n)\neq 1$. Show all positive integers occur in the sequence.

Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$.

Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.

A representation of $\frac{17}{20}$ as a sum of reciprocals $$ \frac{17}{20} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} $$ is called a [i]calm representation[/i] with $k$ terms if the $a_i$ are distinct positive integers and at most one of them is not a power of two. (a) Find the smallest value of $k$ for which $\frac{17}{20}$ has a calm representation with $k$ terms. (b) Prove that there are infinitely many calm representations of $\frac{17}{20}$.

Given a sequence of positive integers $a_1,a_2,a_3,...$ defined by $a_n=\lfloor n^{\frac{2018}{2017}}\rfloor$. Show that there exists a positive integer $N$ such that among any $N$ consecutive terms in the sequence, there exists a term whose decimal representation contain digit $5$.

Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if \[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\] for every prime number $p$ and every real number $y.$

Positive integers $1, 2, 3, ..., n$ are written on a blackboard ($n > 2$). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number $97$ remains. Find the least $n$ for which it is possible.

Let $p$ be a prime number greater than 5. Suppose there is an integer $k$ satisfying that $k^2+5$ is divisible by $p$. Prove that there are positive integers $m$ and $n$ such that $p^2=m^2+5n^2$

Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? [i](Proposed by Richard Henner)[/i]

Let $k$ be a positive integer. A positive integer $n$ is called a $k$-good number if it satisfies the following two conditions: 1. $n$ has exactly $2k$ digits in decimal representation (it cannot have leading zeros). 2. If the first $k$ digits and the last $k$ digits of $n$ are considered as two separate $k$-digit numbers (which may have leading zeros), the square of their sum is equal to $n$. For example, $2025$ is a $2$-good number because \[(20 + 25)^2 = 2025.\] Find all $3$-good numbers.

Find all positive integers that are equal to $700$ times the sum of its digits.

On a board there are $n\geq2$ distinct nonnegative integers such that the sum of each two distinct numbers is a power of $2$. What are the possible values of $n$?

The sequence of natural numbers $1, 5, 6, 25, 26, 30, 31,...$ is made up of powers of $5$ with natural exponents or sums of powers of $5$ with different natural exponents, written in ascending order. Determine the term of the string written in position $167$.

Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$

How many $8$-digit numbers in base $4$ formed of the digits $1,2, 3$ are divisible by $3$?

Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime to $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let \[S=\sum_{d|2008}\phi(d),\] in which $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S$ is divided by $1000$.

Let $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Find all $p$ such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum. [i]A. Khrabov[/i]

A square is drawn in the Cartesian coordinate plane with vertices at $(2,2)$, $(-2,2)$, $(-2,-2)$, and $(2,-2)$. A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $\frac{1}{8}$ that the particle will move from $(x,y)$ to each of $(x,y+1)$, $(x+1,y+1)$, $(x+1,y)$, $(x+1,y-1)$, $(x,y-1)$, $(x-1,y-1)$, $(x-1,y)$, $(x-1,y+1)$. The particle will eventually hit the square for the first time, either at one of the $4$ corners of the square or one of the $12$ lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 39$

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Positive integers $a,b,c$ satisfy the equations $a^2=b^3+ab$ and $c^3=a+b+c$. Prove that $a=bc$.

Compute the remainder when the product of all positive integers less than and relatively prime to $2019$ is divided by $2019$.

The positive integers $a_0, a_1, a_2, \ldots, a_{3030}$ satisfy $$2a_{n + 2} = a_{n + 1} + 4a_n \text{ for } n = 0, 1, 2, \ldots, 3028.$$ Prove that at least one of the numbers $a_0, a_1, a_2, \ldots, a_{3030}$ is divisible by $2^{2020}$.

Find all positive integers $ x,y,z,t$ such that $ x,y,z$ are pairwise coprime and $ (x \plus{} y)(y \plus{} z)(z \plus{} x) \equal{} xyzt$.