We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ such that $n$ has a multiple which is alternating.

Consider a square $(p-1)\times(p-1)$, where $p$ is a prime number, which is divided by squares $1\times 1$ whose sides are parallel to the initial square's sides. Show that it is possible to select $p$ vertices such that there are no three collinear vertices.

Show that for any natural number n, n^3 + (n + 1)^3 + (n + 2)^3 is divisible by 9.

Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

Let $\mathbb R$ be the real numbers. Set $S = \{1, -1\}$ and define a function $\operatorname{sign} : \mathbb R \to S$ by \[ \operatorname{sign} (x) = \begin{cases} 1 & \text{if } x \ge 0; \\ -1 & \text{if } x < 0. \end{cases} \] Fix an odd integer $n$. Determine whether one can find $n^2+n$ real numbers $a_{ij}, b_i \in S$ (here $1 \le i, j \le n$) with the following property: Suppose we take any choice of $x_1, x_2, \dots, x_n \in S$ and consider the values \begin{align*} y_i &= \operatorname{sign} \left( \sum_{j=1}^n a_{ij} x_j \right), \quad \forall 1 \le i \le n; \\ z &= \operatorname{sign} \left( \sum_{i=1}^n y_i b_i \right) \end{align*} Then $z=x_1 x_2 \dots x_n$.

Let $ n$ be a positive integer, and let $ \left( a_{1},\ a_{2} ,\ ...,\ a_{n}\right)$, $ \left( b_{1},\ b_{2},\ ...,\ b_{n}\right)$ and $ \left( c_{1},\ c_{2},\ ...,\ c_{n}\right)$ be three sequences of integers such that for any two distinct numbers $ i$ and $ j$ from the set $ \left\{ 1,2,...,n\right\}$, none of the seven integers $ a_{i}\minus{}a_{j}$; $ \left( b_{i}\plus{}c_{i}\right) \minus{}\left( b_{j}\plus{}c_{j}\right)$; $ b_{i}\minus{}b_{j}$; $ \left( c_{i}\plus{}a_{i}\right) \minus{}\left( c_{j}\plus{}a_{j}\right)$; $ c_{i}\minus{}c_{j}$; $ \left( a_{i}\plus{}b_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\right)$; $ \left( a_{i}\plus{}b_{i}\plus{}c_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\plus{}c_{j}\right)$ is divisible by $ n$. Prove that: [b]a)[/b] The number $ n$ is odd. [b]b)[/b] The number $ n$ is not divisible by $ 3$. [hide="Source of the problem"][i]Source of the problem:[/i] This question is a generalization of one direction of Theorem 2.1 in: Dean Alvis, Michael Kinyon, [i]Birkhoff's Theorem for Panstochastic Matrices[/i], American Mathematical Monthly, 1/2001 (Vol. 108), pp. 28-37. The original Theorem 2.1 is obtained if you require $ b_{i}\equal{}i$ and $ c_{i}\equal{}\minus{}i$ for all $ i$, and add in a converse stating that such sequences $ \left( a_{1},\ a_{2},\ ...,\ a_{n}\right)$, $ \left( b_{1},\ b_{2},\ ...,\ b_{n}\right)$ and $ \left( c_{1} ,\ c_{2},\ ...,\ c_{n}\right)$ indeed exist if $ n$ is odd and not divisible by $ 3$.[/hide]

Let $ A\equal{}\{n: 1 \le n \le 2009^{2009},n \in \mathbb{N} \}$ and let $ S\equal{}\{n: n \in A,\gcd \left(n,2009^{2009}\right)\equal{}1\}$. Let $ P$ be the product of all elements of $ S$. Prove that \[ P \equiv 1 \pmod{2009^{2009}}.\] [i]Nanang Susyanto, Jogjakarta[/i]

Let $ S\equal{}\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x\plus{}y\in A$ or $ x\plus{}y\minus{}n\in A$. Show that the number of elements of $ A$ divides $ n$.

A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased? $\textbf{(A) } 6\qquad \textbf{(B) }7 \qquad \textbf{(C) }8 \qquad \textbf{(D) } 9\qquad \textbf{(E) }\text{cannot be determined}$

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

A magician asked a spectator to think of a three-digit number $ \overline{abc}$ and then to tell him the sum of numbers $ \overline{acb}$, $ \overline{bac}$, $ \overline{bca}$, $ \overline{cab}$, and $ \overline{cba}$. He claims that when he knows this sum he can determine the original number. Is that so?

Let $k,n>1$ be integers such that the number $p=2k-1$ is prime. Prove that, if the number $\binom{n}{2}-\binom{k}{2}$ is divisible by $p$, then it is divisible by $p^2$.

Let $n$ be the smallest positive integer such that the remainder of $3n+45$, when divided by $1060$, is $16$. Find the remainder of $18n+17$ upon division by $1920$.

Let $p$ be a prime such that $p \equiv 3 \pmod 4$. Prove that we can't partition the numbers $a,a+1,a+2,\cdots,a+p-2$,($a \in \mathbb Z$) in two sets such that product of members of the sets be equal.

Consider the triangular array of numbers with $0,1,2,3,...$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows $1$ through $6$ are shown. \begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\ & & & & 1 & & 1 & & & & \\ & & & 2 & & 2 & & 2 & & & \\ & & 3 & & 4 & & 4 & & 3 & & \\ & 4 & & 7 & & 8 & & 7 & & 4 & \\ 5 & & 11 & & 15 & & 15 & & 11 & & 5 \end{tabular} Let $f(n)$ denote the sum of the numbers in row $n$. What is the remainder when $f(100)$ is divided by $100$? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 74$

Let be a group $ G $ of order $ 1+p, $ where $ p $ is and odd prime. Show that if $ p $ divides the number of automorphisms of $ G, $ then $ p\equiv 3\pmod 4. $

Seven students count from $1$ to $1000$ as follows: [list] [*]Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1, 3, 4, 6, 7, 9, \cdots, 997, 999, 1000.$ [*]Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers. [*]Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers. [*]Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers. [*]Finally, George says the only number that no one else says. [/list] What number does George say? $ \textbf{(A)}\ 37\qquad\textbf{(B)}\ 242\qquad\textbf{(C)}\ 365\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 998 $

Find all integer solutions to $2 x^4 + 1 = y^2.$

Does there exist a positive integer $ n>1$ such that $ n$ is a power of $ 2$ and one of the numbers obtained by permuting its (decimal) digits is a power of $ 3$ ?

Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers.

Show that the equation $x^7 + y^7 = {1998}^z$ has no solution in positive integers.

Let $q$ and $r$ be integers with $q>0,$ and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B,$ and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T.$ Show that if the product of the lengths of $A$ and $B$ is less than $q,$ then $S$ is the intersection of $A$ with some arithmetic progression.

Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$

Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

There is a pile of eggs. Joan counted the eggs, but her count was way off by $1$ in the $1$'s place. Tom counted in the eggs, but his count was off by $1$ in the $10$'s place. Raoul counted the eggs, but his count was off by $1$ in the $100$'s place. Sasha, Jose, Peter, and Morris all counted the eggs and got the correct count. When these seven people added their counts together, the sum was $3162$. How many eggs were in the pile?