In decimal representation, we call an integer [i]$k$-carboxylic[/i] if and only if it can be represented as a sum of $k$ distinct integers, all of them greater than $9$, whose digits are the same. For instance, $2008$ is [i]$5$-carboxylic[/i] because $2008=1111+666+99+88+44$. Find, with an example, the smallest integer $k$ such that $8002$ is [i]$k$-carboxylic[/i].

Show that there exists a polynomial $P(x)$ with integer coefficients, such that the equation $P(x) = 0$ has no integer solutions, but for each positive integer $n$ there is an $x \in \Bbb{Z}$ such that $n \mid P(x).$

Find the least positive integer $n$ such that $15$ divides the product \[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\] , for every positive integers $a_1, a_2, \dots, a_{15}$.

For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be [i]amusing[/i] if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than $1$?

Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $1,2,\ldots,n^2$ as digits. That is, the equation\[ r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0 \]is true for a unique choice of non-negative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from the set $\{0,1,2,\ldots,n^2\}$, with $a_m\ne 0$. We write \[ r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i} \]to denote the base $-n+i$ expansion of $r+si$. There are only finitely many integers $k+0i$ that have four-digit expansions \[ k=(a_3a_2a_1a_0)_{-3+i}~~~~a_3\ne 0. \]Find the sum of all such $k$.

Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$. [list=a] [*] Determine its units digit. [*] Determine its tens digit. [/list]

Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.

For a natural number $n$, set $S_n$ is defined as: \[S_n = \left \{ {n\choose n}, {2n \choose n}, {3n\choose n},..., {n^2 \choose n} \right \}.\] a) Prove that there are infinitely many composite numbers $n$, such that the set $S_n$ is not complete residue system mod $n$; b) Prove that there are infinitely many composite numbers $n$, such that the set $S_n$ is complete residue system mod $n$.

Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.

The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Suppose $p$ is a prime with $p \equiv 3 \; \pmod{4}$. Show that for any set of $p-1$ consecutive integers, the set cannot be divided two subsets so that the product of the members of the one set is equal to the product of the members of the other set.

The digits of the three-digit integers $a, b,$ and $c$ are the nine nonzero digits $1,2,3,\cdots 9$ each of them appearing exactly once. Given that the ratio $a:b:c$ is $1:3:5$, determine $a, b,$ and $c$.

The sequence $\{a_{n}\}_{n\geq 0}$ is defined by $a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...)$. Find the smallest positive integer $h$ satisfying $1998|a_{n+h}-a_{n}\forall n=0,1,2,...$

It is given that for some $n \in \mathbb{N}$ there exists a natural number $a$, such that $a^{n-1} \equiv 1 \pmod{n}$ and that for any prime divisor $p$ of $n-1$ we have $a^{\frac{n-1}{p}} \not \equiv 1 \pmod{n}$. Prove that $n$ is a prime.

Let $\,n>6\,$ be an integer and $\,a_{1},a_{2},\ldots,a_{k}\,$ be all the natural numbers less than $n$ and relatively prime to $n$. If \[a_{2}-a_{1}=a_{3}-a_{2}=\cdots =a_{k}-a_{k-1}>0,\] prove that $\,n\,$ must be either a prime number or a power of $\,2$.

(a) Is $2005^{2004}$ the sum of two perfect squares? (b) Is $2004^{2005}$ the sum of two perfect squares?

Problem. The sum of two consecutive squares can be a square; for instance $3^2 + 4^2 = 5^2$. (a) Prove that the sum of $m$ consecutive squares cannot be a square for $m \in \{3, 4, 5, 6\}$. (b) Find an example of eleven consecutive squares whose sum is a square. Can anyone help me with this? Thanks.

Find all natural numbers such that \[ n\sigma(n)\equiv 2\pmod {\phi( n)}\]

Let $a$ be an odd positive integer greater than 17 such that $3a-2$ is a perfect square. Show that there exist distinct positive integers $b$ and $c$ such that $a+b,a+c,b+c$ and $a+b+c$ are four perfect squares.

Let $p\ge5$ be a prime. Show that \[\sum_{k=0}^{(p-1)/2}\binom{p}{k}3^k\equiv 2^p - 1\pmod{p^2}.\] [i]Victor Wang.[/i]

Let $ A\equal{}\{1,2,\ldots, 2008\}$. We will say that set $ X$ is an $ r$-set if $ \emptyset \neq X \subset A$, and $ \sum_{x\in X} x \equiv r \pmod 3$. Let $ X_r$, $ r\in\{0,1,2\}$ be the set of $ r$-sets. Find which one of $ X_r$ has the most elements.

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

Find the smallest positive integer whose cube ends in 888.

How many ordered integer pairs of $ (a,b)$ satisfying $ a^2b \plus{} ab^2 \equal{} 2009201020092010$ ? $\textbf{(A)} 4 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None}$

Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-[i]limited[/i] if $ |\sigma(i)\minus{}i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k\plus{}1}.$