How many different solutions does the congruence $x^3+3x^2+x+3 \equiv 0 \pmod{25}$ have? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6 $

The lengths of the sides of a triangle are integers, and its area is also an integer. One side is $21$ and the perimeter is $48$. The shortest side is: $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list]

Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

Find the number of primes$ p$, such that $ x^{3} \minus{}5x^{2} \minus{}22x\plus{}56\equiv 0\, \, \left(mod\, p\right)$ has no three distinct integer roots in $ \left[0,\left. p\right)\right.$. $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$

Prove that the number of ordered triples $(x, y, z)$ such that $(x+y+z)^2 \equiv axyz \mod{p}$, where $gcd(a, p) = 1$ and $p$ is prime is $p^2 + 1$.

Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals.

Let \(n\) be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these \(n\) sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by \(n\)?

A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is $ 1$ place to its right in the alphabet (assuming that the letter $ A$ is one place to the right of the letter $ Z$). The second time this same letter appears in the given message, it is replaced by the letter that is $ 1\plus{}2$ places to the right, the third time it is replaced by the letter that is $ 1 \plus{} 2 \plus{} 3$ places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter $ \text{s}$ in the message "Lee's sis is a Mississippi miss, Chriss!"? $ \textbf{(A)}\ \text{g}\qquad \textbf{(B)}\ \text{h}\qquad \textbf{(C)}\ \text{o}\qquad \textbf{(D)}\ \text{s}\qquad \textbf{(E)}\ \text{t}$

The expression $\circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder? [i]Proposed by Aaron Lin[/i]

Let $\overline{a_1a_2a_3\ldots a_n}$ be the representation of a $n-$digits number in base $10.$ Prove that there exists a one-to-one function like $f : \{0, 1, 2, 3, \ldots, 9\} \to \{0, 1, 2, 3, \ldots, 9\}$ such that $f(a_1) \neq 0$ and the number $\overline{ f(a_1)f(a_2)f(a_3) \ldots f(a_n) }$ is divisible by $3.$

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

Prove that if $m=5^n+3^n+1$ is a prime, then $12$ divides $n$.

Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]

Let $S$ be equal to the sum \[1+2+3+\cdots+2007.\] Find the remainder when $S$ is divided by $1000$.

For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers. Prove that: 1. There does not exist a solution $(a, b, c)$ for $n = 2017$. 2. For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$. 3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$.

A sequence of positive integers is defined by $a_0=1$ and $a_{n+1}=a_n^2+1$ for each $n\ge0$. Find $\text{gcd}(a_{999},a_{2004})$.

On a $2004 \times 2004$ chess table there are 2004 queens such that no two are attacking each other\footnote[1]{two queens attack each other if they lie on the same row, column or direction parallel with on of the main diagonals of the table}. Prove that there exist two queens such that in the rectangle in which the center of the squares on which the queens lie are two opposite corners, has a semiperimeter of 2004.

a) Show that the set $ \mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $ A,B,C$ satisfying the following conditions: \[ BA = B; \& B^2 = C; \& BC = A; \] where $ HK$ stands for the set $ \{hk: h \in H, k \in K\}$ for any two subsets $ H, K$ of $ \mathbb{Q}^{ + }$ and $ H^2$ stands for $ HH.$ b) Show that all positive rational cubes are in $ A$ for such a partition of $ \mathbb{Q}^{ + }.$ c) Find such a partition $ \mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $ n \leq 34,$ both $ n$ and $ n + 1$ are in $ A,$ that is, \[ \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34. \]

Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \ldots$ of nonzero integers such that the equality \[a_k+2a_{2k}+\ldots+na_{nk}=0\]holds for every positive integer $k$.

Suppose that $ p$ is a prime number. Prove that the equation $ x^2\minus{}py^2\equal{}\minus{}1$ has a solution if and only if $ p\equiv1\pmod 4$.

Consider a sequence of positive integers $a_1, a_2, a_3, . . .$ such that for $k \geq 2$ we have $a_{k+1} =\frac{a_k + a_{k-1}}{2015^i},$ where $2015^i$ is the maximal power of $2015$ that divides $a_k + a_{k-1}.$ Prove that if this sequence is periodic then its period is divisible by $3.$

In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

Let $a,b$ be two integers. Prove that a) $13 \mid 2a+3b$ if and only if $13 \mid 2b-3a$; b) If $13 \mid a^2+b^2$ then $13 \mid (2a+3b)(2b+3a)$. [i]Mircea Fianu[/i]

Let $ \clubsuit(x)$ denote the sum of the digits of the positive integer $ x$. For example, $ \clubsuit(8)\equal{}8$ and $ \clubsuit(123)\equal{}1\plus{}2\plus{}3\equal{}6$. For how many two-digit values of $ x$ is $ \clubsuit(\clubsuit(x))\equal{}3$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$